Page 1 Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 POLYNOMIALS "The word polynomial means an algebraic expression consisting of one or many terms involving powers of the variable. Poly means many or much. INTRODUCTION In our earlier classes, we have studied about algebraic expressions, their addition, subtraction and multiplication etc. We have also studied about factorisation using some formulae. We have also learnt the formulae. (x ± y) 2 = x 2 ± 2xy + y 2 x 2 – y 2 = (x + y) (x – y) (x + a) (x + b) = x 2 + (a + b)x + ab x 3 – y 3 = (x – y) (x 2 + xy + y 2 ) x 3 + y 3 = (x + y) (x 2 – xy + y 2 ) and their use in factorisation. In this chapter we shall study about particular types of algebraic expressions, called polynomials. We shall also study the remainder and factor theorems and shall use them in factorisation of polynomials. We shall also discuss some more algebraic indentities and their use in factorisation. HISTORICAL FACTS Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou". We would write 3x + 2y + z = 29. R ene Descartes The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, – for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithmetica Integra, 1544. Rene Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen in the general formula for a polynomial, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. CONSTANT AND A VARIABLE In algebra, two types of symbols are used : constants and variables (literals) ( i ) Constant : A symbol whose value remains same throughout a particular problem is called constant. Ex. : 4, –9, 3 8 , ?, 7 15 etc. ( i i ) Variable : A symbol whose value changes according to the situation is called variable or literal. It is denoted by the letters of alphabet a, b, c ....... x, y, z, etc. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. ALGEBRAIC EXPRESSIONS A combination of constants and variables connected by any fundamental operation (i.e. +, –, × and ÷) is called an algebraic expression. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. Page 2 Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 POLYNOMIALS "The word polynomial means an algebraic expression consisting of one or many terms involving powers of the variable. Poly means many or much. INTRODUCTION In our earlier classes, we have studied about algebraic expressions, their addition, subtraction and multiplication etc. We have also studied about factorisation using some formulae. We have also learnt the formulae. (x ± y) 2 = x 2 ± 2xy + y 2 x 2 – y 2 = (x + y) (x – y) (x + a) (x + b) = x 2 + (a + b)x + ab x 3 – y 3 = (x – y) (x 2 + xy + y 2 ) x 3 + y 3 = (x + y) (x 2 – xy + y 2 ) and their use in factorisation. In this chapter we shall study about particular types of algebraic expressions, called polynomials. We shall also study the remainder and factor theorems and shall use them in factorisation of polynomials. We shall also discuss some more algebraic indentities and their use in factorisation. HISTORICAL FACTS Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou". We would write 3x + 2y + z = 29. R ene Descartes The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, – for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithmetica Integra, 1544. Rene Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen in the general formula for a polynomial, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. CONSTANT AND A VARIABLE In algebra, two types of symbols are used : constants and variables (literals) ( i ) Constant : A symbol whose value remains same throughout a particular problem is called constant. Ex. : 4, –9, 3 8 , ?, 7 15 etc. ( i i ) Variable : A symbol whose value changes according to the situation is called variable or literal. It is denoted by the letters of alphabet a, b, c ....... x, y, z, etc. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. ALGEBRAIC EXPRESSIONS A combination of constants and variables connected by any fundamental operation (i.e. +, –, × and ÷) is called an algebraic expression. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 TERM OF AN EXPRESSION A constant, a variable or a combination of a constant and variable connected with the operation of multiplication or division is called a term. Algebraic expression Number of terms Terms – 25x 1 – 25x 2x + 3y 2 2x and 3y 4x 2 – 3xy + 5z 3 4x 2 , – 3xy, 5z 3 y xy 2 x 5 7 ? ? ? 4 3 y xy , , and 2 x 5 7 ? Factors of term : The algebraic expression 4x 2 + 5xy 2 consists of two terms 4x 2 and 5xy 2 . The term 4x 2 is the product of 4, x and x and the factors of the term 5xy 2 are 5, x, y and y. So, we can express the factors of 4x 2 + 5xy 2 in terms of tree diagram as follows :- 4x + 5xy 2 2 4 x x 4x 2 + 5xy 2 5 x y y COEFFICIENTS Any factor of a term is called the coefficient of the remaining term. For example :- (i) In 7x ; 7 is coefficient of x (ii) In – 5x 2 y ; 5 is coefficient of –x 2 y ; –5 is coefficient of x 2 y, Ex. Write the coefficient of : (i) x 2 in 3x 3 – 5x 2 + 7 (ii) xy in 8 xyz (iii) –y in 2y 2 – 6y + 2 (iv) x 0 in 3x + 7 Sol. (i) –5 (ii) 8z (iii) 6 (iv) Since x 0 = 1, Therefore 3x + 7 = 3x + 7x 0 ? coefficient of x 0 is 7. ( i ) POLYNOMIALS An algebraic expression of the form p(x) = a n x n + a n–1 x n–1 + a n–2 x n–2 +............+ a 1 x 1 + a 0 x 0 where a n ? 0 and a 0 , a 1 , a 2 ,.......a n are real numbers and each power of x is a positive integer, is called a polynomial. Hence, a n , a n–1 , a n–2 , are coefficients of x n , x n–1 ..............x 0 and a n x n , a n–1 x n–1 , a n–2 x n–2 ,............ are terms of the polynomial. Here the term a n x n is called the leading term and its coefficient a n , the leading coefficient. For example : p(u) = 1 2 u 3 – 3u 2 + 2u – 4 is a polynomial in variable u. 1 2 u 3 , –3u 2 , 2u, –4 are known as terms of polynomial and 1 2 , –3, 2, –4 are their respective coefficients. 6x –2 This is NOT a polynomial term Because the variable has a negative exponent This is NOT a polynomial term Because the variable is in the denominator sqrt(x) This is NOT a polynomial term Because the variable is inside a radical 4x 2 This is a polynomial term Because it obeys all the rules 2 1 x Page 3 Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 POLYNOMIALS "The word polynomial means an algebraic expression consisting of one or many terms involving powers of the variable. Poly means many or much. INTRODUCTION In our earlier classes, we have studied about algebraic expressions, their addition, subtraction and multiplication etc. We have also studied about factorisation using some formulae. We have also learnt the formulae. (x ± y) 2 = x 2 ± 2xy + y 2 x 2 – y 2 = (x + y) (x – y) (x + a) (x + b) = x 2 + (a + b)x + ab x 3 – y 3 = (x – y) (x 2 + xy + y 2 ) x 3 + y 3 = (x + y) (x 2 – xy + y 2 ) and their use in factorisation. In this chapter we shall study about particular types of algebraic expressions, called polynomials. We shall also study the remainder and factor theorems and shall use them in factorisation of polynomials. We shall also discuss some more algebraic indentities and their use in factorisation. HISTORICAL FACTS Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou". We would write 3x + 2y + z = 29. R ene Descartes The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, – for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithmetica Integra, 1544. Rene Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen in the general formula for a polynomial, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. CONSTANT AND A VARIABLE In algebra, two types of symbols are used : constants and variables (literals) ( i ) Constant : A symbol whose value remains same throughout a particular problem is called constant. Ex. : 4, –9, 3 8 , ?, 7 15 etc. ( i i ) Variable : A symbol whose value changes according to the situation is called variable or literal. It is denoted by the letters of alphabet a, b, c ....... x, y, z, etc. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. ALGEBRAIC EXPRESSIONS A combination of constants and variables connected by any fundamental operation (i.e. +, –, × and ÷) is called an algebraic expression. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 TERM OF AN EXPRESSION A constant, a variable or a combination of a constant and variable connected with the operation of multiplication or division is called a term. Algebraic expression Number of terms Terms – 25x 1 – 25x 2x + 3y 2 2x and 3y 4x 2 – 3xy + 5z 3 4x 2 , – 3xy, 5z 3 y xy 2 x 5 7 ? ? ? 4 3 y xy , , and 2 x 5 7 ? Factors of term : The algebraic expression 4x 2 + 5xy 2 consists of two terms 4x 2 and 5xy 2 . The term 4x 2 is the product of 4, x and x and the factors of the term 5xy 2 are 5, x, y and y. So, we can express the factors of 4x 2 + 5xy 2 in terms of tree diagram as follows :- 4x + 5xy 2 2 4 x x 4x 2 + 5xy 2 5 x y y COEFFICIENTS Any factor of a term is called the coefficient of the remaining term. For example :- (i) In 7x ; 7 is coefficient of x (ii) In – 5x 2 y ; 5 is coefficient of –x 2 y ; –5 is coefficient of x 2 y, Ex. Write the coefficient of : (i) x 2 in 3x 3 – 5x 2 + 7 (ii) xy in 8 xyz (iii) –y in 2y 2 – 6y + 2 (iv) x 0 in 3x + 7 Sol. (i) –5 (ii) 8z (iii) 6 (iv) Since x 0 = 1, Therefore 3x + 7 = 3x + 7x 0 ? coefficient of x 0 is 7. ( i ) POLYNOMIALS An algebraic expression of the form p(x) = a n x n + a n–1 x n–1 + a n–2 x n–2 +............+ a 1 x 1 + a 0 x 0 where a n ? 0 and a 0 , a 1 , a 2 ,.......a n are real numbers and each power of x is a positive integer, is called a polynomial. Hence, a n , a n–1 , a n–2 , are coefficients of x n , x n–1 ..............x 0 and a n x n , a n–1 x n–1 , a n–2 x n–2 ,............ are terms of the polynomial. Here the term a n x n is called the leading term and its coefficient a n , the leading coefficient. For example : p(u) = 1 2 u 3 – 3u 2 + 2u – 4 is a polynomial in variable u. 1 2 u 3 , –3u 2 , 2u, –4 are known as terms of polynomial and 1 2 , –3, 2, –4 are their respective coefficients. 6x –2 This is NOT a polynomial term Because the variable has a negative exponent This is NOT a polynomial term Because the variable is in the denominator sqrt(x) This is NOT a polynomial term Because the variable is inside a radical 4x 2 This is a polynomial term Because it obeys all the rules 2 1 x Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 4 ( i i ) TYPES OF POLYNOMIALS Generally we divide the polynomials in three categories. T ypes of Polynomials Based on number of term s Based on number of distinct variables Based on degree Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s Na m e Exa m ple 1 Univa ria te x + 9 2 Biva ria te x + y + 9 3 Triva ria te x + y + z + 9 Generally, a polynomial in more than one variable is called a multivariate polynomial. A second major way of classifying polynomials is by their degree. Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term. Polynomials classified by degree D egree Name Example – ? Zero 0 0 (non-zero) constant 1 1 Linear x + 1 2 quadratic x 2 + 1 3 cubic x 3 + 2 4 quartic (or biquadratic) x 4 + 3 5 quintic x 5 + 4 6 sextic (or hexic) x 6 + 5 7 septic (or heptic) x 7 + 6 8 octic x 8 + 7 9 nonic x 9 + 8 10 decic x 10 + 9 Usually, a polynomial of degree n, for n greater than 3, is called a polynomial of degree n, although the phrases quartic polynomial and quintic polynomial are sometimes used. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined to be negative (either –1 or – ?) N um be r o f no n -z e ro te rm s N a m e Exa m ple 0 ze ro p olynom ia l 0 1 m onom ia l x 2 2 binom ia l x 2 + 1 3 trinom ia l x 2 + x + 1 P o lyno m ia ls c la s s ifie d by num be r o f no n-z e ro te rm s If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). Page 4 Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 POLYNOMIALS "The word polynomial means an algebraic expression consisting of one or many terms involving powers of the variable. Poly means many or much. INTRODUCTION In our earlier classes, we have studied about algebraic expressions, their addition, subtraction and multiplication etc. We have also studied about factorisation using some formulae. We have also learnt the formulae. (x ± y) 2 = x 2 ± 2xy + y 2 x 2 – y 2 = (x + y) (x – y) (x + a) (x + b) = x 2 + (a + b)x + ab x 3 – y 3 = (x – y) (x 2 + xy + y 2 ) x 3 + y 3 = (x + y) (x 2 – xy + y 2 ) and their use in factorisation. In this chapter we shall study about particular types of algebraic expressions, called polynomials. We shall also study the remainder and factor theorems and shall use them in factorisation of polynomials. We shall also discuss some more algebraic indentities and their use in factorisation. HISTORICAL FACTS Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou". We would write 3x + 2y + z = 29. R ene Descartes The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, – for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithmetica Integra, 1544. Rene Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen in the general formula for a polynomial, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. CONSTANT AND A VARIABLE In algebra, two types of symbols are used : constants and variables (literals) ( i ) Constant : A symbol whose value remains same throughout a particular problem is called constant. Ex. : 4, –9, 3 8 , ?, 7 15 etc. ( i i ) Variable : A symbol whose value changes according to the situation is called variable or literal. It is denoted by the letters of alphabet a, b, c ....... x, y, z, etc. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. ALGEBRAIC EXPRESSIONS A combination of constants and variables connected by any fundamental operation (i.e. +, –, × and ÷) is called an algebraic expression. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 TERM OF AN EXPRESSION A constant, a variable or a combination of a constant and variable connected with the operation of multiplication or division is called a term. Algebraic expression Number of terms Terms – 25x 1 – 25x 2x + 3y 2 2x and 3y 4x 2 – 3xy + 5z 3 4x 2 , – 3xy, 5z 3 y xy 2 x 5 7 ? ? ? 4 3 y xy , , and 2 x 5 7 ? Factors of term : The algebraic expression 4x 2 + 5xy 2 consists of two terms 4x 2 and 5xy 2 . The term 4x 2 is the product of 4, x and x and the factors of the term 5xy 2 are 5, x, y and y. So, we can express the factors of 4x 2 + 5xy 2 in terms of tree diagram as follows :- 4x + 5xy 2 2 4 x x 4x 2 + 5xy 2 5 x y y COEFFICIENTS Any factor of a term is called the coefficient of the remaining term. For example :- (i) In 7x ; 7 is coefficient of x (ii) In – 5x 2 y ; 5 is coefficient of –x 2 y ; –5 is coefficient of x 2 y, Ex. Write the coefficient of : (i) x 2 in 3x 3 – 5x 2 + 7 (ii) xy in 8 xyz (iii) –y in 2y 2 – 6y + 2 (iv) x 0 in 3x + 7 Sol. (i) –5 (ii) 8z (iii) 6 (iv) Since x 0 = 1, Therefore 3x + 7 = 3x + 7x 0 ? coefficient of x 0 is 7. ( i ) POLYNOMIALS An algebraic expression of the form p(x) = a n x n + a n–1 x n–1 + a n–2 x n–2 +............+ a 1 x 1 + a 0 x 0 where a n ? 0 and a 0 , a 1 , a 2 ,.......a n are real numbers and each power of x is a positive integer, is called a polynomial. Hence, a n , a n–1 , a n–2 , are coefficients of x n , x n–1 ..............x 0 and a n x n , a n–1 x n–1 , a n–2 x n–2 ,............ are terms of the polynomial. Here the term a n x n is called the leading term and its coefficient a n , the leading coefficient. For example : p(u) = 1 2 u 3 – 3u 2 + 2u – 4 is a polynomial in variable u. 1 2 u 3 , –3u 2 , 2u, –4 are known as terms of polynomial and 1 2 , –3, 2, –4 are their respective coefficients. 6x –2 This is NOT a polynomial term Because the variable has a negative exponent This is NOT a polynomial term Because the variable is in the denominator sqrt(x) This is NOT a polynomial term Because the variable is inside a radical 4x 2 This is a polynomial term Because it obeys all the rules 2 1 x Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 4 ( i i ) TYPES OF POLYNOMIALS Generally we divide the polynomials in three categories. T ypes of Polynomials Based on number of term s Based on number of distinct variables Based on degree Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s Na m e Exa m ple 1 Univa ria te x + 9 2 Biva ria te x + y + 9 3 Triva ria te x + y + z + 9 Generally, a polynomial in more than one variable is called a multivariate polynomial. A second major way of classifying polynomials is by their degree. Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term. Polynomials classified by degree D egree Name Example – ? Zero 0 0 (non-zero) constant 1 1 Linear x + 1 2 quadratic x 2 + 1 3 cubic x 3 + 2 4 quartic (or biquadratic) x 4 + 3 5 quintic x 5 + 4 6 sextic (or hexic) x 6 + 5 7 septic (or heptic) x 7 + 6 8 octic x 8 + 7 9 nonic x 9 + 8 10 decic x 10 + 9 Usually, a polynomial of degree n, for n greater than 3, is called a polynomial of degree n, although the phrases quartic polynomial and quintic polynomial are sometimes used. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined to be negative (either –1 or – ?) N um be r o f no n -z e ro te rm s N a m e Exa m ple 0 ze ro p olynom ia l 0 1 m onom ia l x 2 2 binom ia l x 2 + 1 3 trinom ia l x 2 + x + 1 P o lyno m ia ls c la s s ifie d by num be r o f no n-z e ro te rm s If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 4 ( i i i ) VALUE OF A POLYNOMIAL Let f(x) be a polynomial in variable x and ? is any real number, then the value of polynomial at x = ? is obtained by substituting x = ? in the value of polynomial and is denoted by f( ?). Ex. : Consider a polynomial f(x) = 3x 2 – 4x + 2, find the value at x = 3. replace x by 3 everywhere So, the value of f(x) = 3x 2 – 4x + 2 at x = 3 is f(3) = 3 × (3) 2 – 4 × 3 + 2 = 27 – 12 + 2 = 17 Similarly the value of polynomial f(x) = 3x 2 – 4x + 2 at x = –2 is f(–2) = 3(–2) 2 – 4 × (–2) + 2 = 12 + 8 + 2 = 22 at x = 0 is f(0) = 3(0) 2 – 4 (0) + 2 = 0 – 0 + 2 = 2 at x = 1 2 is f ? ? 1 2 = 3 × ? ? 2 1 2 – 4 × ? ? 1 2 + 2 = 3 4 – 2 + 2 = 3 4 DO Y OU R SE L F Find the value of polynomial 5x – 4x 2 + 3 at : (i) x = 0 (ii) x = –1 (iii) x = 2 ( i v ) ZEROES OF A POLYNOMIAL A real number ? is a zero of a polynomial p(x) if the value of the polynomial p(x) is zero at x = ?. i.e. p( ?) = 0 O R The value of the variable x, for which the polynomial p(x) becomes zero is called zero of the polynomial. Ex. : consider, a polynomial p(x) = x 2 – 5x + 6 ; replace x by 2 and 3. p(2) = (2) 2 – 5 × 2 + 6 = 4 – 10 + 6 = 0, p(3) = (3) 2 – 5 × 3 + 6 = 9 – 15 + 6 = 0 ? 2 and 3 are the zeroes of the polynomial p(x). R E M A R K 1. The constant polynomial has no zero. 2. Every linear polynomial has one and only one zero or root. consider a linear polynomial p(x) = ax + b, a ? 0 ? p(x) = 0 ? ax + b = 0 ? ax = –b ? x = – b a is a zero of the polynomial. 3. A given polynomial can have more than one zero or root. 4. If the degree of a polynomial is n, the maximum number of zeroes it can have is also n. Ex. : If the degree of a polynomial is 5, the polynomial can have at the most 5 zeroes, if the degree of polynomial is 8, maximum number of zeroes it can have is 8. 5. A zero of a polynomial need not be 0 Ex. : Consider a polynomial f(x) = x 2 – 4, then f(2) = (2) 2 – 4 = 4 – 4 = 0 here, zero of the polynomial f(x) = x 2 – 4 is 2 which itself is not 0. 6. 0 may be zero of the polynomial. Ex. : Consider a polynomial f(x) = x 2 – x, then f(0) = 0 2 – 0 = 0 here, 0 is the zero of the polynomial f(x) = x 2 – x Page 5 Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 POLYNOMIALS "The word polynomial means an algebraic expression consisting of one or many terms involving powers of the variable. Poly means many or much. INTRODUCTION In our earlier classes, we have studied about algebraic expressions, their addition, subtraction and multiplication etc. We have also studied about factorisation using some formulae. We have also learnt the formulae. (x ± y) 2 = x 2 ± 2xy + y 2 x 2 – y 2 = (x + y) (x – y) (x + a) (x + b) = x 2 + (a + b)x + ab x 3 – y 3 = (x – y) (x 2 + xy + y 2 ) x 3 + y 3 = (x + y) (x 2 – xy + y 2 ) and their use in factorisation. In this chapter we shall study about particular types of algebraic expressions, called polynomials. We shall also study the remainder and factor theorems and shall use them in factorisation of polynomials. We shall also discuss some more algebraic indentities and their use in factorisation. HISTORICAL FACTS Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou". We would write 3x + 2y + z = 29. R ene Descartes The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, – for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithmetica Integra, 1544. Rene Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen in the general formula for a polynomial, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. CONSTANT AND A VARIABLE In algebra, two types of symbols are used : constants and variables (literals) ( i ) Constant : A symbol whose value remains same throughout a particular problem is called constant. Ex. : 4, –9, 3 8 , ?, 7 15 etc. ( i i ) Variable : A symbol whose value changes according to the situation is called variable or literal. It is denoted by the letters of alphabet a, b, c ....... x, y, z, etc. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. ALGEBRAIC EXPRESSIONS A combination of constants and variables connected by any fundamental operation (i.e. +, –, × and ÷) is called an algebraic expression. Ex. : 3x + 5y, 7y – 2x, 2x – ay + az etc. Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 2 TERM OF AN EXPRESSION A constant, a variable or a combination of a constant and variable connected with the operation of multiplication or division is called a term. Algebraic expression Number of terms Terms – 25x 1 – 25x 2x + 3y 2 2x and 3y 4x 2 – 3xy + 5z 3 4x 2 , – 3xy, 5z 3 y xy 2 x 5 7 ? ? ? 4 3 y xy , , and 2 x 5 7 ? Factors of term : The algebraic expression 4x 2 + 5xy 2 consists of two terms 4x 2 and 5xy 2 . The term 4x 2 is the product of 4, x and x and the factors of the term 5xy 2 are 5, x, y and y. So, we can express the factors of 4x 2 + 5xy 2 in terms of tree diagram as follows :- 4x + 5xy 2 2 4 x x 4x 2 + 5xy 2 5 x y y COEFFICIENTS Any factor of a term is called the coefficient of the remaining term. For example :- (i) In 7x ; 7 is coefficient of x (ii) In – 5x 2 y ; 5 is coefficient of –x 2 y ; –5 is coefficient of x 2 y, Ex. Write the coefficient of : (i) x 2 in 3x 3 – 5x 2 + 7 (ii) xy in 8 xyz (iii) –y in 2y 2 – 6y + 2 (iv) x 0 in 3x + 7 Sol. (i) –5 (ii) 8z (iii) 6 (iv) Since x 0 = 1, Therefore 3x + 7 = 3x + 7x 0 ? coefficient of x 0 is 7. ( i ) POLYNOMIALS An algebraic expression of the form p(x) = a n x n + a n–1 x n–1 + a n–2 x n–2 +............+ a 1 x 1 + a 0 x 0 where a n ? 0 and a 0 , a 1 , a 2 ,.......a n are real numbers and each power of x is a positive integer, is called a polynomial. Hence, a n , a n–1 , a n–2 , are coefficients of x n , x n–1 ..............x 0 and a n x n , a n–1 x n–1 , a n–2 x n–2 ,............ are terms of the polynomial. Here the term a n x n is called the leading term and its coefficient a n , the leading coefficient. For example : p(u) = 1 2 u 3 – 3u 2 + 2u – 4 is a polynomial in variable u. 1 2 u 3 , –3u 2 , 2u, –4 are known as terms of polynomial and 1 2 , –3, 2, –4 are their respective coefficients. 6x –2 This is NOT a polynomial term Because the variable has a negative exponent This is NOT a polynomial term Because the variable is in the denominator sqrt(x) This is NOT a polynomial term Because the variable is inside a radical 4x 2 This is a polynomial term Because it obeys all the rules 2 1 x Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 4 ( i i ) TYPES OF POLYNOMIALS Generally we divide the polynomials in three categories. T ypes of Polynomials Based on number of term s Based on number of distinct variables Based on degree Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s Na m e Exa m ple 1 Univa ria te x + 9 2 Biva ria te x + y + 9 3 Triva ria te x + y + z + 9 Generally, a polynomial in more than one variable is called a multivariate polynomial. A second major way of classifying polynomials is by their degree. Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term. Polynomials classified by degree D egree Name Example – ? Zero 0 0 (non-zero) constant 1 1 Linear x + 1 2 quadratic x 2 + 1 3 cubic x 3 + 2 4 quartic (or biquadratic) x 4 + 3 5 quintic x 5 + 4 6 sextic (or hexic) x 6 + 5 7 septic (or heptic) x 7 + 6 8 octic x 8 + 7 9 nonic x 9 + 8 10 decic x 10 + 9 Usually, a polynomial of degree n, for n greater than 3, is called a polynomial of degree n, although the phrases quartic polynomial and quintic polynomial are sometimes used. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined to be negative (either –1 or – ?) N um be r o f no n -z e ro te rm s N a m e Exa m ple 0 ze ro p olynom ia l 0 1 m onom ia l x 2 2 binom ia l x 2 + 1 3 trinom ia l x 2 + x + 1 P o lyno m ia ls c la s s ifie d by num be r o f no n-z e ro te rm s If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 4 ( i i i ) VALUE OF A POLYNOMIAL Let f(x) be a polynomial in variable x and ? is any real number, then the value of polynomial at x = ? is obtained by substituting x = ? in the value of polynomial and is denoted by f( ?). Ex. : Consider a polynomial f(x) = 3x 2 – 4x + 2, find the value at x = 3. replace x by 3 everywhere So, the value of f(x) = 3x 2 – 4x + 2 at x = 3 is f(3) = 3 × (3) 2 – 4 × 3 + 2 = 27 – 12 + 2 = 17 Similarly the value of polynomial f(x) = 3x 2 – 4x + 2 at x = –2 is f(–2) = 3(–2) 2 – 4 × (–2) + 2 = 12 + 8 + 2 = 22 at x = 0 is f(0) = 3(0) 2 – 4 (0) + 2 = 0 – 0 + 2 = 2 at x = 1 2 is f ? ? 1 2 = 3 × ? ? 2 1 2 – 4 × ? ? 1 2 + 2 = 3 4 – 2 + 2 = 3 4 DO Y OU R SE L F Find the value of polynomial 5x – 4x 2 + 3 at : (i) x = 0 (ii) x = –1 (iii) x = 2 ( i v ) ZEROES OF A POLYNOMIAL A real number ? is a zero of a polynomial p(x) if the value of the polynomial p(x) is zero at x = ?. i.e. p( ?) = 0 O R The value of the variable x, for which the polynomial p(x) becomes zero is called zero of the polynomial. Ex. : consider, a polynomial p(x) = x 2 – 5x + 6 ; replace x by 2 and 3. p(2) = (2) 2 – 5 × 2 + 6 = 4 – 10 + 6 = 0, p(3) = (3) 2 – 5 × 3 + 6 = 9 – 15 + 6 = 0 ? 2 and 3 are the zeroes of the polynomial p(x). R E M A R K 1. The constant polynomial has no zero. 2. Every linear polynomial has one and only one zero or root. consider a linear polynomial p(x) = ax + b, a ? 0 ? p(x) = 0 ? ax + b = 0 ? ax = –b ? x = – b a is a zero of the polynomial. 3. A given polynomial can have more than one zero or root. 4. If the degree of a polynomial is n, the maximum number of zeroes it can have is also n. Ex. : If the degree of a polynomial is 5, the polynomial can have at the most 5 zeroes, if the degree of polynomial is 8, maximum number of zeroes it can have is 8. 5. A zero of a polynomial need not be 0 Ex. : Consider a polynomial f(x) = x 2 – 4, then f(2) = (2) 2 – 4 = 4 – 4 = 0 here, zero of the polynomial f(x) = x 2 – 4 is 2 which itself is not 0. 6. 0 may be zero of the polynomial. Ex. : Consider a polynomial f(x) = x 2 – x, then f(0) = 0 2 – 0 = 0 here, 0 is the zero of the polynomial f(x) = x 2 – x Polynomianls – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 5 6 Ex. Find which of the following algebraic expression is a polynomial. (i) 3x 2 – 5x (ii) x + 1 x (iii) y – 8 (iv) z 5 – 3 z + 8 Sol. ( i ) 3x 2 – 5x = 3x 2 – 5x 1 Is a polynomial in one variable. ( i i ) x + 1 x = x 1 + x –1 Is not a polynomial as the second term 1/x has degree (–1). ( i i i ) y – 8 = y 1/2 – 8 Is not a polynomial since, the power of the first term ( y) is 1 2 , which is not a whole number. ( i v ) z 5 – 3 z + 8 = z 5 – z 1/3 + 8 Since, the exponent of the second term is 1/3, which in not a whole number. Therefore, the given expression is not a polynomial. Ex. Find the degree of the polynomial : (i) 5x 2 – 6x 3 + 8x 7 + 6x 2 (ii) 2y 12 + 3y 10 – y 15 + y + 3 (iii) x (iv) 8 Sol. ( i ) Since the term with highest exponent (power) is 8x 7 and its power is 7. ? The degree of given polynomial is 7. ( i i ) The highest power of the variable is 15. ? degree = 15 ( i i i )x = x 1 ? degree is 1. ( i v ) 8 = 8x 0 ? degree is 0. Ex. Write the coefficient of x 2 in each of the following : (i) 2 + x 2 + x (ii) 5 – x 2 + x 3 (iii) 1 2 x 2 + x (iv) 5 x – 1 Sol. ( i ) In the polynomial 2 + x 2 + x the coefficient of x 2 is 1. ( i i ) In the polynomial 5 – x 2 + x 3 the coefficient of x 2 is – 1. ( i i i )In the polynomial 1 2 x 2 + x the coefficient of x 2 is 1 2 . ( i v ) In the polynomial 5 x – 1 the coefficient of x 2 = 0. [We may write 5 x – 1 = ax 2 + 5 x – 1. So the coefficient of x 2 is 0] Ex. Classify the following as linear, quadratic and cubic polynomials : (i) x 2 + x (ii) x – x 3 (iii) 1 + x (iv) 3t (v) r 2 (vi) 7x 3 (vii) y + y 2 + 5 (viii) 3xyz Sol. ( i ) The polynomial x 2 + x is a quadratic polynomial as its degree is 2. ( i i ) Degree of the polynomial x – x 3 is 3. It is a cubic polynomial ( i i i )Degree of the polynomial 1 + x is 1. It is a linear polynomial. ( i v ) Degree of the polynomial 3t is 1. It is a linear polynomial. ( v ) Degree of the polynomial r 2 is 2. It is a quadratic polynomial. ( v i ) Degree of the polynomial 7x 3 is 3. It is a cubic polynomial. ( v i i )Degree of the polynomial y + y 2 + 5 is 2. It is a quadratic polynomial. ( v i i i ) 3xyz is a polynomial in 3 variables x, y and z. Its degree is 1 + 1 + 1 = 3. It is a cubic polynomial. Ex. Find q(0), q(1) and q(2) for each of the following polynomials : (i) q(x) = x 2 + 3x (ii) q(y) = 2 + y + 2y 2 – 5y 3 (iii) q(t) = t 3Read More

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