Page 1 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 1 POLYNOMIALS ? INTRODUCTION In class IX, we have studied the polynomials in one variable and their degrees. We have also learnt about the values and the zeros of a polynomial. In this chapter, we wil discuss more about the zeros of a polynomial and the relationship between the zeros and the coefficients of a polynomial with particular reference to quadratic polynomials. In addition, statement and simple problems on division algorithm for polynomials with real coefficients will be discussed. ? 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. 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. ? RECALL ( 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. 6 x – 2 This is N O T a p olynom ia l te rm Be ca use the va ria ble ha s a ne ga tive e xp one nt This is N O T a p olynom ia l te rm Be ca use the va ria ble is in the de nom ina tor sqrt(x) This is N O T a p olynom ia l te rm Be ca use the va ria ble is inside a ra dica l 4 x 2 This IS a p olynom ia l te rm Be ca use it obe ys a ll the rule s 2 1 x ( i i ) Types of Polynomials : Generally we divide the polynomials in three categories. T ypes of Polynom ials Based on number of term s Based on number of distinct variables Based on degree R ene Descartes Page 2 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 1 POLYNOMIALS ? INTRODUCTION In class IX, we have studied the polynomials in one variable and their degrees. We have also learnt about the values and the zeros of a polynomial. In this chapter, we wil discuss more about the zeros of a polynomial and the relationship between the zeros and the coefficients of a polynomial with particular reference to quadratic polynomials. In addition, statement and simple problems on division algorithm for polynomials with real coefficients will be discussed. ? 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. 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. ? RECALL ( 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. 6 x – 2 This is N O T a p olynom ia l te rm Be ca use the va ria ble ha s a ne ga tive e xp one nt This is N O T a p olynom ia l te rm Be ca use the va ria ble is in the de nom ina tor sqrt(x) This is N O T a p olynom ia l te rm Be ca use the va ria ble is inside a ra dica l 4 x 2 This IS a p olynom ia l te rm Be ca use it obe ys a ll the rule s 2 1 x ( i i ) Types of Polynomials : Generally we divide the polynomials in three categories. T ypes of Polynom ials Based on number of term s Based on number of distinct variables Based on degree R ene Descartes Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 2 Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s N a m e Exa m ple 1 U niva 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 Degree 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"). ( i i i ) Value of a Polynomial : If p(x) is a polynomial in variable x and ? is any real number, then the value obtained by replacing x by ? ?in p(x) is called value of p(x) at x = ? and is denoted by p( ?). For example : Find the value of p(x) = x 3 – 6 x 2 + 11x – 6 at x = –2 ? p(–2) = (–2) 3 – 6 (–2) 2 + 11(–2) – 6 = –8 – 24 – 22 – 6 ? p(–2) = – 60 Page 3 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 1 POLYNOMIALS ? INTRODUCTION In class IX, we have studied the polynomials in one variable and their degrees. We have also learnt about the values and the zeros of a polynomial. In this chapter, we wil discuss more about the zeros of a polynomial and the relationship between the zeros and the coefficients of a polynomial with particular reference to quadratic polynomials. In addition, statement and simple problems on division algorithm for polynomials with real coefficients will be discussed. ? 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. 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. ? RECALL ( 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. 6 x – 2 This is N O T a p olynom ia l te rm Be ca use the va ria ble ha s a ne ga tive e xp one nt This is N O T a p olynom ia l te rm Be ca use the va ria ble is in the de nom ina tor sqrt(x) This is N O T a p olynom ia l te rm Be ca use the va ria ble is inside a ra dica l 4 x 2 This IS a p olynom ia l te rm Be ca use it obe ys a ll the rule s 2 1 x ( i i ) Types of Polynomials : Generally we divide the polynomials in three categories. T ypes of Polynom ials Based on number of term s Based on number of distinct variables Based on degree R ene Descartes Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 2 Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s N a m e Exa m ple 1 U niva 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 Degree 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"). ( i i i ) Value of a Polynomial : If p(x) is a polynomial in variable x and ? is any real number, then the value obtained by replacing x by ? ?in p(x) is called value of p(x) at x = ? and is denoted by p( ?). For example : Find the value of p(x) = x 3 – 6 x 2 + 11x – 6 at x = –2 ? p(–2) = (–2) 3 – 6 (–2) 2 + 11(–2) – 6 = –8 – 24 – 22 – 6 ? p(–2) = – 60 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 3 ( i v ) Zero of a Polynomial : A real number ? is a zero of the polynomial p(x) if p( ?) = 0. For example : Consider p(x) = x 3 – 6x 2 + 11x – 6 p(1) = (1) 3 – 6(1) 2 + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 p(2) = (2) 3 – 6(2) 2 + 11(2) – 6 = 8 – 24 + 22 – 6 = 0 p(3) = (3) 3 – 6(3) 2 + 11(3) – 6 = 27 – 54 + 33 – 6 = 0 Thus, 1,2 and 3 are called the zeros of polynomial p(x). ? GEOMETRICAL MEANING OF THE ZEROS OF A POLYNOMIAL Geometrically the zeros of a polynomials f(x) are the x-co-ordinates of the points where the graph y = f(x) intersects x-axis. To understand it, we will see the geometrical representations of linear and quadratic polynomials. Geometrical Representation of the zero of a Linear Polynomial Consider a linear polynomial, y = 2x – 5. The following table lists the values of y corresponding to different values of x. x 1 4 y – 3 3 On plotting the points A(1, –3) and B(4, 3) and joining them, a straight line is obtained. B ( 4,3) P( ,0) y x' y ' x 5 2 ? ? A ( 1, –3) O ? From, graph we observe that the graph of y = 2x – 5 intersects the x-axis at 5 , 0 2 ? ? ? ? ? ? whose x-coordinate is 5 2 . Also, zero of 2x – 5 is 5 2 . Therefore, we conclude that the linear polynomial ax + b has one and only one zero, which is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis Geometrical Representation of the zero of a Quadratic Polynomial : Consider a quadratic polynomial, y = x 2 – 2x – 8, The following table gives the values of y or f(x) for various values of x. x –4 –3 –2 –1 0 1 2 3 4 5 6 y = x 2 – 2 x – 8 1 6 7 0 -5 –8 –9 –8 –5 0 7 1 6 On plotting the points (–4, 16), (–3, 7) (–2, 0), (–1, –5), (0, –8), (1, –9) (2, –8), (3, –5), (4, 0), (5, 7) and (6, 16) on a graph paper and drawing a smooth free hand curve passing through these points, the curve thus obtained represents the graph of the polynomial y = x 2 – 2x – 8. This is called a parabola. Page 4 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 1 POLYNOMIALS ? INTRODUCTION In class IX, we have studied the polynomials in one variable and their degrees. We have also learnt about the values and the zeros of a polynomial. In this chapter, we wil discuss more about the zeros of a polynomial and the relationship between the zeros and the coefficients of a polynomial with particular reference to quadratic polynomials. In addition, statement and simple problems on division algorithm for polynomials with real coefficients will be discussed. ? 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. 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. ? RECALL ( 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. 6 x – 2 This is N O T a p olynom ia l te rm Be ca use the va ria ble ha s a ne ga tive e xp one nt This is N O T a p olynom ia l te rm Be ca use the va ria ble is in the de nom ina tor sqrt(x) This is N O T a p olynom ia l te rm Be ca use the va ria ble is inside a ra dica l 4 x 2 This IS a p olynom ia l te rm Be ca use it obe ys a ll the rule s 2 1 x ( i i ) Types of Polynomials : Generally we divide the polynomials in three categories. T ypes of Polynom ials Based on number of term s Based on number of distinct variables Based on degree R ene Descartes Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 2 Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s N a m e Exa m ple 1 U niva 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 Degree 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"). ( i i i ) Value of a Polynomial : If p(x) is a polynomial in variable x and ? is any real number, then the value obtained by replacing x by ? ?in p(x) is called value of p(x) at x = ? and is denoted by p( ?). For example : Find the value of p(x) = x 3 – 6 x 2 + 11x – 6 at x = –2 ? p(–2) = (–2) 3 – 6 (–2) 2 + 11(–2) – 6 = –8 – 24 – 22 – 6 ? p(–2) = – 60 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 3 ( i v ) Zero of a Polynomial : A real number ? is a zero of the polynomial p(x) if p( ?) = 0. For example : Consider p(x) = x 3 – 6x 2 + 11x – 6 p(1) = (1) 3 – 6(1) 2 + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 p(2) = (2) 3 – 6(2) 2 + 11(2) – 6 = 8 – 24 + 22 – 6 = 0 p(3) = (3) 3 – 6(3) 2 + 11(3) – 6 = 27 – 54 + 33 – 6 = 0 Thus, 1,2 and 3 are called the zeros of polynomial p(x). ? GEOMETRICAL MEANING OF THE ZEROS OF A POLYNOMIAL Geometrically the zeros of a polynomials f(x) are the x-co-ordinates of the points where the graph y = f(x) intersects x-axis. To understand it, we will see the geometrical representations of linear and quadratic polynomials. Geometrical Representation of the zero of a Linear Polynomial Consider a linear polynomial, y = 2x – 5. The following table lists the values of y corresponding to different values of x. x 1 4 y – 3 3 On plotting the points A(1, –3) and B(4, 3) and joining them, a straight line is obtained. B ( 4,3) P( ,0) y x' y ' x 5 2 ? ? A ( 1, –3) O ? From, graph we observe that the graph of y = 2x – 5 intersects the x-axis at 5 , 0 2 ? ? ? ? ? ? whose x-coordinate is 5 2 . Also, zero of 2x – 5 is 5 2 . Therefore, we conclude that the linear polynomial ax + b has one and only one zero, which is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis Geometrical Representation of the zero of a Quadratic Polynomial : Consider a quadratic polynomial, y = x 2 – 2x – 8, The following table gives the values of y or f(x) for various values of x. x –4 –3 –2 –1 0 1 2 3 4 5 6 y = x 2 – 2 x – 8 1 6 7 0 -5 –8 –9 –8 –5 0 7 1 6 On plotting the points (–4, 16), (–3, 7) (–2, 0), (–1, –5), (0, –8), (1, –9) (2, –8), (3, –5), (4, 0), (5, 7) and (6, 16) on a graph paper and drawing a smooth free hand curve passing through these points, the curve thus obtained represents the graph of the polynomial y = x 2 – 2x – 8. This is called a parabola. Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 4 y x' y ' x ? 0 –6 –4 –1 –5 –2 –3 1 3 2 5 6 4 –2 –4 –6 –8 2 4 6 8 10 12 14 16 ? ? ? ? ? ? ? ? ? y = x – 2x – 8 2 ( –4, 16) ( –3, 7) ( –2, 0) ( –1, – 5) ( 0, –8) ( 2, –8) ( 3, –5) ( 4, 0) ( 5, 7) ( 6, 16) It is clear from the table that –2 and 4 are the zeros of the quadratic polynomial x 2 – 2x – 8. Also, we observe that – 2 and 4 are the x-coordinates of the points where the graph of y = x 2 – 2x – 8 intersects the x-axis. Consider the following cases – Case-I : Here, the graph cuts x-axis at two distinct points A and A'. The x-coordinates of A and A' are two zeroes of the quadratic polynomial ax 2 + bx + c. y y ' x' x y =ax+bx+c 2 ( i) A A ' y y ' x' x y =ax+bx+c 2 ( ii) A A ' Case-II : Here, the graph cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A' of Case (i) coincide here to become one point A. y y ' x' x y =ax+bx+c 2 ( iii) A ? O y y ' x' x y =ax+bx +c 2 ( iv ) A A ' O The x-coordinate of A is the only zero for the quadratic polynomial ax 2 + bx + c in this case. Page 5 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 1 POLYNOMIALS ? INTRODUCTION In class IX, we have studied the polynomials in one variable and their degrees. We have also learnt about the values and the zeros of a polynomial. In this chapter, we wil discuss more about the zeros of a polynomial and the relationship between the zeros and the coefficients of a polynomial with particular reference to quadratic polynomials. In addition, statement and simple problems on division algorithm for polynomials with real coefficients will be discussed. ? 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. 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. ? RECALL ( 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. 6 x – 2 This is N O T a p olynom ia l te rm Be ca use the va ria ble ha s a ne ga tive e xp one nt This is N O T a p olynom ia l te rm Be ca use the va ria ble is in the de nom ina tor sqrt(x) This is N O T a p olynom ia l te rm Be ca use the va ria ble is inside a ra dica l 4 x 2 This IS a p olynom ia l te rm Be ca use it obe ys a ll the rule s 2 1 x ( i i ) Types of Polynomials : Generally we divide the polynomials in three categories. T ypes of Polynom ials Based on number of term s Based on number of distinct variables Based on degree R ene Descartes Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 2 Polynomials classified by number of distinct variables Num be r o f dis tinc t va ria ble s N a m e Exa m ple 1 U niva 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 Degree 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"). ( i i i ) Value of a Polynomial : If p(x) is a polynomial in variable x and ? is any real number, then the value obtained by replacing x by ? ?in p(x) is called value of p(x) at x = ? and is denoted by p( ?). For example : Find the value of p(x) = x 3 – 6 x 2 + 11x – 6 at x = –2 ? p(–2) = (–2) 3 – 6 (–2) 2 + 11(–2) – 6 = –8 – 24 – 22 – 6 ? p(–2) = – 60 Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 3 ( i v ) Zero of a Polynomial : A real number ? is a zero of the polynomial p(x) if p( ?) = 0. For example : Consider p(x) = x 3 – 6x 2 + 11x – 6 p(1) = (1) 3 – 6(1) 2 + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 p(2) = (2) 3 – 6(2) 2 + 11(2) – 6 = 8 – 24 + 22 – 6 = 0 p(3) = (3) 3 – 6(3) 2 + 11(3) – 6 = 27 – 54 + 33 – 6 = 0 Thus, 1,2 and 3 are called the zeros of polynomial p(x). ? GEOMETRICAL MEANING OF THE ZEROS OF A POLYNOMIAL Geometrically the zeros of a polynomials f(x) are the x-co-ordinates of the points where the graph y = f(x) intersects x-axis. To understand it, we will see the geometrical representations of linear and quadratic polynomials. Geometrical Representation of the zero of a Linear Polynomial Consider a linear polynomial, y = 2x – 5. The following table lists the values of y corresponding to different values of x. x 1 4 y – 3 3 On plotting the points A(1, –3) and B(4, 3) and joining them, a straight line is obtained. B ( 4,3) P( ,0) y x' y ' x 5 2 ? ? A ( 1, –3) O ? From, graph we observe that the graph of y = 2x – 5 intersects the x-axis at 5 , 0 2 ? ? ? ? ? ? whose x-coordinate is 5 2 . Also, zero of 2x – 5 is 5 2 . Therefore, we conclude that the linear polynomial ax + b has one and only one zero, which is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis Geometrical Representation of the zero of a Quadratic Polynomial : Consider a quadratic polynomial, y = x 2 – 2x – 8, The following table gives the values of y or f(x) for various values of x. x –4 –3 –2 –1 0 1 2 3 4 5 6 y = x 2 – 2 x – 8 1 6 7 0 -5 –8 –9 –8 –5 0 7 1 6 On plotting the points (–4, 16), (–3, 7) (–2, 0), (–1, –5), (0, –8), (1, –9) (2, –8), (3, –5), (4, 0), (5, 7) and (6, 16) on a graph paper and drawing a smooth free hand curve passing through these points, the curve thus obtained represents the graph of the polynomial y = x 2 – 2x – 8. This is called a parabola. Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 4 y x' y ' x ? 0 –6 –4 –1 –5 –2 –3 1 3 2 5 6 4 –2 –4 –6 –8 2 4 6 8 10 12 14 16 ? ? ? ? ? ? ? ? ? y = x – 2x – 8 2 ( –4, 16) ( –3, 7) ( –2, 0) ( –1, – 5) ( 0, –8) ( 2, –8) ( 3, –5) ( 4, 0) ( 5, 7) ( 6, 16) It is clear from the table that –2 and 4 are the zeros of the quadratic polynomial x 2 – 2x – 8. Also, we observe that – 2 and 4 are the x-coordinates of the points where the graph of y = x 2 – 2x – 8 intersects the x-axis. Consider the following cases – Case-I : Here, the graph cuts x-axis at two distinct points A and A'. The x-coordinates of A and A' are two zeroes of the quadratic polynomial ax 2 + bx + c. y y ' x' x y =ax+bx+c 2 ( i) A A ' y y ' x' x y =ax+bx+c 2 ( ii) A A ' Case-II : Here, the graph cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A' of Case (i) coincide here to become one point A. y y ' x' x y =ax+bx+c 2 ( iii) A ? O y y ' x' x y =ax+bx +c 2 ( iv ) A A ' O The x-coordinate of A is the only zero for the quadratic polynomial ax 2 + bx + c in this case. Polynomials – Aadhar TYCRP 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi | 011-32044009 3 5 Case-III : Here, the graph is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point. y y ' x' x y =ax+bx+c 2 ( v ) O y y ' x' x y =ax+bx+c 2 ( v i) O So, the quadratic polynomial ax 2 + bx +c has no zero in this case. So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or one zero, or no zero. This also means that a polynomial of degree 2 has atmost two zeroes. Remark : In general given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has atmost n zeros. ? RELATIONSHIP BETWEEN THE ZEROS AND COEFFICIENTS OF A POLYNOMIAL For a linear polynomial ax + b, (a ? ?0), we have, b (constant term) zero of a linear polynomial a (coefficient of x) ? ? ? ? For a quadratic polynomial ax 2 + b + c (a ? ?0), with ? ?and ? ?as it's zeros, we have 2 2 b (coefficient of x) Sum of zeros a (coefficient of x ) c (constant term) Product of zeros a (coefficient of x ) ? ? ? ? ? ? ? ? ? ? ? ? ? If ? and ? ?are the zeros of a quadratic polynomial f(x). Then polynomial f(x) is given by f(x) = K{x 2 – ( ? + ?)x + ? ?} or f(x) = K{x 2 – (sum of the zeros) x + product of the zeros} where K is a constant. COMPETITION WINDOW RELATIONSHIP BETWEEN THE ZEROS AND COEFFICIENTS OF A CUBIC POLYNOMIAL For a cubic polynomial ax 3 + bx 2 + cx + d (a ? 0), with ? ? ? ? and ? as its zeros, we have : Sum of three zeros = ? ? ? ? ? ? ? ? ? = –b a Sum of the product of its zeros taken two at a time = ? ? + ? ? + ? ? = c a Product of its zeros = ? ? ? = –d a The cubic polynomial whose zeros are ? ? ? ? and ? is given by f(x) = {x 3 – ( ? ? ? ? ? ? ? ? ?) x 2 + ( ? ? + ? ? + ? ?) x – ? ? ?}Read More

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