Polynomials & Zeroes of a Polynomial Notes | Study Mathematics (Maths) Class 9 - Class 9

Introduction

• Polynomial is made up of two terms, namely Poly (meaning “many”) and Nomial (meaning “terms”). 
• A polynomial is defined as an expression that is composed of variables, constants, and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication, and division. (Division operation by a variable is not allowed).
• Polynomial is an algebraic expression, that has only the whole number powers of the variables.
• In an expression, generally, we have variables and constant terms.
  Example:  3x² + 2x + 6; Here x is variable and 6 without any variable is called constant.
• The polynomial function is denoted by P(x) where x represents the variable. 
  Example: P(x) = x² - 5x + 11; If the variable is denoted by a, then the function will be P(a)
• The terms of polynomials are the parts of the equation that are generally separated by “+” or “-” signs. So, each part of a polynomial in an equation is a term. 
  Example: In a polynomial, say, 2x² + 5x + 4, the number of terms will be 3.

 Degree of a Polynomial

• The degree of an algebraic equation is the highest degree for a term with a non-zero coefficient.
• The degree of a term is the sum of the powers of each variable in the term.
  Examples:
  (i) 4x⁷-5
  In this expression, we have only one term and the power of the term is 7, so the expression has a degree, 7.
  (ii) 5 x³y³ - 4 x²y² + 7xy
  Here, it has 3 terms, and the power of the first term is 3+3=6, the power of the second term is 2+2=4, and the power of the last term is 1+1=2, here the highest power is 6, so the expression has a degree, 3+3=6.

We can do basic arithmetic operations with two algebraic equation. That is we can add, subtract, multiply or divide any two polynomial.

Polynomial in One Variable

• The degree of polynomials in one variable is the highest power of the variable in the algebraic expression.
• Example: In the following equation: x²+2x+4. The degree of the equation is 2, .i.e. the highest power of variable in the equation.

Multivariable Polynomial

• For a multivariable polynomial, it the highest sum of powers of different variables in any of the terms in the expression.
  Example:  x⁵+3x⁴y+2xy³+4y²-2y+1.  
  It is a multivariable polynomial in x and y, and the degree of the polynomial is 5 – as you can see, the degree in the terms x⁵ is 5, x⁴y is also 5 (4+1) and so the highest degree among these individual terms is 5.
• A polynomial of two variable, x and y, like ax^ry^s is the algebraic sum of several terms of the prior mentioned form, where r and s are possible integers. Here, the degree of the polynomial is r+s where r and s are whole numbers.

Note: Exponents of variables of a polynomial .i.e. degree of polynomials should be whole numbers.

How to find the Degree of a Polynomial?

There are 4 simple steps are present to find the degree of a polynomial:

Example: 6x⁵+8x³+3x⁵+3x²+4+2x+4 

• Step 1: Combine all the like terms that are the terms of the variable terms.(6x⁵+3x⁵)+8x³+3x²+2x+(4+4) 
• Step 2: Ignore all the coefficients.
  x⁵+x³+x²+x+x⁰ 
• Step 3: Arrange the variable in descending order of their powers.
  x⁵+x³+x²+x+x⁰ 
• Step 4: The largest power of the variable is the degree of the polynomial.
  deg(x⁵+x³+x²+x+x⁰) = 5 

Properties of Polynomial

Some of the important properties of polynomials along with some important polynomial theorems are as follows:

(i) Division Algorithm

If a polynomial P(x) is divided by a polynomial G(x) results in quotient Q(x) with remainder R(x), then, P(x) = G(x) • Q(x) + R(x)

(ii) Bezout’s Theorem

Polynomial P(x) is divisible by binomial (x – a) if and only if P(a) = 0.

(iii) Remainder Theorem

If P(x) is divided by (x – a) with remainder r, then P(a) = r.

(iv) Factor Theorem

A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).

(v) Intermediate Value Theorem

If P(x) is a polynomial, and P(x) ≠ P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].

(vi) The addition, subtraction and multiplication of polynomials P and Q result in a polynomial where,

Degree(P ± Q) ≤ Degree(P or Q)

Degree(P × Q) = Degree(P) + Degree(Q)

(vii) If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.

(viii) If a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by (Q • R).

(ix) If P(x) = a₀ + a₁x + a₂x² + …… + a_nxⁿ is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.

(x) Descartes’ Rule of Sign

The number of positive real zeroes in a polynomial function P(x) is the same or less than by an even number as the number of changes in the sign of the coefficients. So, if there are “K” sign changes, the number of roots will be “k” or “(k – a)”, where “a” is some even number.

(xi) Fundamental Theorem of Algebra

Every non-constant single-variable polynomial with complex coefficients has at least one complex root.

(xii) If P(x) is a polynomial with real coefficients and has one complex zero (x = a – bi), then x = a + bi will also be a zero of P(x). Also, x² – 2ax + a² + b² will be a factor of P(x).

Arithmetic Operations 

1. Addition

For adding any two polynomials, we have to combine the like terms.
Example: Add 4 x² + 7 x - 6, x² - 3 x + 2
∵ Like terms are 4x² and x², 7x and -3x, -6 and 2
⇒ If we combine 4x² and x², we will get 5x²
⇒ If we combine 7x and -3x, we will get 4x
⇒ If we combine -6 and 2, we will get -4
Thus, the final answer is 5x² + 4x - 4

2. Subtraction

(2 x³ - 2 x² + 4 x - 3)- (x³ + x² - 5 x + 4)
Step 1: Multiply the negative with inner terms i.e. 2x³ -2x² + 4x -3 - x³-x²+5x-4
Step 2: Combine the like terms i.e. 2x³ - x³ -2x²-x² + 4x + 5x - 3 - 4
Step 3: The answer i.e. x³ + x² + 9x - 7

3. Multiplication

There are two formats for this: horizontal and vertical, like in addition.

The simplest case of the multiplication of polynomials is the multiplication of monomials.
Example: (5x²)(- 2x³)
For multiplying these two monomials, we have to just multiply the numbers and add the powers using the exponent rule.
⇒  (5x²)(-2x³) = -10x² ⁺ ³= -10x⁵

4. Division

Division of polynomials involves two cases, the first one is a simplification, which is reducing the fraction and the second one is long division.

Zeroes of a Polynomial

If the value of a polynomial is zero for some value of the variable then that value is known as zero of the polynomial.

Definition

Let p(x) be a polynomial in x. If p(x) = 0, then we say that a is a zero of the polynomial p(x).

Zero of a Linear Polynomial

Example 1. Find the zeros of the following linear polynomial
p(x)  =  2x + 3
Solution: p(x)  =  2x + 3
Now we have to think about the value of x, for which the given function will become zero.
For that let us factor out 2
p(x)  =  2 (x + 3/2)
Instead of "x", if we apply -3/2, p(x) will become zero.
Hence -3/2 is the zero of the given linear polynomial.
Example 2. Find the zeros of the following linear polynomial
p(x)  =  4x - 1
Solution: p(x)  =  4x - 1
Now we have to think about the value of x, for which the given function will become zero.
For that let us factor out 4
p(x)  =  4 (x - 1/4)
By applying the value 1/4 instead of x, the function p(x) will become zero.
Hence 1/4 is the zero of the given linear polynomial.

Zeroes of a Quadratic Polynomial

For a quadratic equation, there will be two zeros. In order to find those zeros, we may use the method called factoring.

Example 3. Find the zeros of the quadratic equation x² + 17 x + 60 by factoring.
Solution: p(x) = x² + 17x + 60
p(x) = x² + 12x + 5x + 60
p(x) = x(x + 12) + 5(x + 12)
p(x) = (x + 5)(x + 12)
If x = -5; p(x) = (-5 + 5)(-5 + 12) = 0
If x = -12; p(x) = (-12 + 5)(-12 + 12) = 0
Hence the zeros are -5 and -12.

Zeroes of a Cubic Polynomial

For a cubic equation, there will be three zeros. 

In order to find those zeroes, we may use the methods:
(i)  Factor theorem: Factor theorem is commonly used for factoring a polynomial and finding the roots of the polynomial. It is a special case of a polynomial remainder theorem.
(ii)  Synthetic division: A simplified method for dividing a polynomial by another polynomial of the first degree by writing down only the coefficients of the several powers of the variable and changing the sign of the constant term in the divisor so as to replace the usual subtractions by additions.

Example 4. Find the zeros of the following polynomial 4 x³ - 7 x + 3
Solution: Let p (x) = 4 x³ - 7 x + 3 x = 1
p (1) = 4 (1)³ -7 (1) + 3
= 4 - 7 + 3
= 7 - 7
= 0
So we can decide (x - 1) is a factor. To find the other two factors, we have to use synthetic division.  

So, the factors are (x - 1) and (4 x² - 4 x - 3). By factorizing this quadratic equation, we get  
(2 x +3) (2 x - 1).
Hence the required factors are (x - 1) (2 x + 3) (2 x - 1).