Math Doubts

Zeros or Roots of a Polynomial

A value for which a polynomial becomes zero is called a zero of a polynomial.

Introduction

The polynomial is equal to zero when a particular value is substituted in the polynomial and that particular value is called a zero of the polynomial. It is also called a root of the polynomial.

Let $f(x)$ be a polynomial in variable $x$ and it is equal to zero for $x$ equals to $\alpha$, $\beta$ and $\gamma$.

  1. $f(\alpha) \,=\, 0$
  2. $f(\beta) \,=\, 0$
  3. $f(\gamma) \,=\, 0$

The values of alpha, beta and gamma made the polynomial $f(x)$ to become zero. Hence, $\alpha$, $\beta$ and $\gamma$ are called the zeros or zeroes of the polynomial, and also called the roots of the polynomial.

Example

Let us understand the concept of a root or zero of a polynomial.

$2x^3$ $-$ $3x^2$ $+$ $7x$ $-$ $6$

It is an example polynomial in one variable. Substitute $x \,=\, 1$ and find the value of the polynomial.

$=\,\,\,$ $2(1)^3$ $-$ $3(1)^2$ $+$ $7(1)$ $-$ $6$

$=\,\,\,$ $2 \times (1)^3$ $-$ $3 \times (1)^2$ $+$ $7 \times (1)$ $-$ $6$

$=\,\,\,$ $2 \times 1^3$ $-$ $3 \times 1^2$ $+$ $7 \times 1$ $-$ $6$

$=\,\,\,$ $2 \times 1$ $-$ $3 \times 1$ $+$ $7 \times 1$ $-$ $6$

$=\,\,\,$ $2$ $-$ $3$ $+$ $7$ $-$ $6$

$=\,\,\,$ $2$ $+$ $7$ $-$ $3$ $-$ $6$

$=\,\,\,$ $9$ $-$ $9$

$=\,\,\,$ $0$

The value of the polynomial $2x^3$ $-$ $3x^2$ $+$ $7x$ $-$ $6$ is equal to zero when the value of $x$ equals to $1$. Therefore, the number $1$ is called a root or zero of the polynomial.

Remember that the degree of a polynomial defines the number of roots or zeros of the polynomial. In this example, the degree of the polynomial is $3$. We know that $x \,=\, 1$ is a root. So, it has two more zeroes.

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