# Polynomials in One Variable – Polynomials | Class 9 Maths

**Polynomial** originated from the words “poly” which means “many” and the word “nomial” which means “term”. In maths, a polynomial expression consists of variables which are also known as **indeterminate** and **coefficients**. Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. In the polynomial, each expression in it is called a term. Suppose x^{2} + 5x + 2 is polynomial, then the expressions x^{2}, 5x, and 2 are the terms of the polynomial. Each term of the polynomial has a coefficient. For example, if 2x + 1 is the polynomial, then the coefficient of x is 2. But in this article we are going to discuss the polynomial in one variable. In general,

The algebraic expressions with only one variable are known as Polynomials in one variable.

**Examples:**

- P(x) = 4x – 3
- G(y) = y
^{4}– y^{2 }+ 2y + 9

**Degree of Polynomials**

The **highest exponent** of the variable in the algebraic expression is called** Degree of a Polynomial.**

**Examples**

- P(x) = x
^{3}– 5x^{2}+ 9x + 16(Degree = 3, because the highest exponent is 3 here)- Q(x) = 5
(Degree = 0)

**Evaluating Polynomials**

The **value of a polynomial at a given value** of the variable is termed as evaluation of polynomial.

**Examples**

**Question 1: Evaluate the polynomial x ^{2} + 3?**

**Solution:**

Let, p(x) = x

^{2}+ 3Value of polynomial at x = 3 will be:

p(3) = (3)

^{2}+ 3=> p(3) = 9 + 3

=> p(3) = 12

**Question 2: Evaluate the polynomial x ^{2} + 2x + 6?**

**Solution:**

Let, q(x) = x

^{2}+ 2x + 6Value of q(x) at x = 0

q(0) = (0)

^{3}+ 2(0) + 6=> q(0) = 6

## Factors of the Polynomial

A factor of a given polynomial (say P(x)) is any polynomial that divides evenly into P(x). The factorization of the polynomial is the representation of the polynomial in the product form.

**Example**

x

^{2}– 4 = (x – 2)(x + 2)

Here, (x – 2) and (x + 2) are the factors of the polynomial x^{2 }– 4 and also at x = 2 and x = -2 the value of polynomial is 0. Therefore, x = 2 and x = -2 are called as **Zeroes of the Polynomial. Zero of the polynomial **is the value of the variable in the polynomial where the value of the Polynomial becomes 0.

**Examples**

1.p(x) = 2x – 4At x = 2 value of polynomial will be 0 and (x – 2) will be factor of p(x).

2.q(x) = (x – 5)^{2}At x = 5 value of polynomial will be 0 and (x – 5) will be a factor of p(x).

**Finding Factors and Zeroes of Polynomial**

Zeroes of the polynomials can be found by equating the given polynomial with Zero(0) and solving the equation for the given variable.

**Examples**

**Question 1. Find Factors and Zeroes of Polynomial f(x) = 5x – 15?**

**Solution:**

Given Polynomial, f(x) = 5x – 15

Now, Equating above polynomial with 0

5x – 15 = 0

=> 5(x – 3) = 0

=> x – 3 =0

=> x = 3

Therefore, x = 3 is the

Zeroof the Polynomial (f(x) = 5x – 15) or theRootof the equation (5x – 15 = 0) and (x – 3) is theFactorof the given polynomial. So, f(x) can be represented as:f(x) = 5(x – 3)

**Question 2. Find Factors and Zeroes of Polynomial f(x) = 2x ^{2} – x – 6?**

**Solution:**

Given Polynomial, f(x) = 2x

^{2}– x – 6Now, Equating above polynomial with 0

2x

^{2}– x – 6 = 0=> 2x

^{2}– 4x + 3x – 6 = 0=> 2x(x – 2) + 3(x – 2) = 0

=> (x – 2)(2x + 3) = 0

So, x – 2 = 0 or 2x + 3 = 0

Therefore, x = 2 and x = -3/2 are the

Zeroesof the Polynomial (f(x) = 2x^{2}– x – 6) or theRootsof the equation (2x^{2}– x – 6 = 0) and (x – 2) & (2x + 3) areFactorsof the given polynomial. So, f(x) can be represented as:f(x) = (x – 2)(2x + 3)

## Graphical Representation of Polynomials

The Polynomials can be represented on the graph paper by plotting it point by point. Let’s see some examples by plotting some graphs on graph paper.

**Examples**

**Question 1. Plot the graph for the polynomial f(x) = 2x.**

**Solution:**

x | 0 | 1 | 2 | 3 | -1 |
---|---|---|---|---|---|

f(x) | 0 | 2 | 4 | 6 | -2 |

**Graph of polynomial: **f(x) = 2x

The point(on the X-axis) where the graph of polynomial cuts the X-axis is called zero of the polynomial.

**Question 2. Plot the graph for the polynomial f(x) = x – 3.**

**Solution:**

x | 0 | 1 | 2 | 3 | 4 | -1 | -2 |
---|---|---|---|---|---|---|---|

f(x) | -3 | -2 | -1 | 0 | 1 | -4 | -5 |

**Graph of polynomial: **f(x) = x – 3

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