# How to Find Zeros of Polynomial?

Zeros of the polynomial are points where the polynomial is equal to zero. Here you will learn how to find the zeros of a polynomial.

Zeros of a polynomial are also known as the roots of the equation and are often denoted by $$α$$, $$β$$, and $$γ$$, respectively. Some of the methods used to find polynomial zeros are grouping, factorization, and the use of algebraic expressions.

## Step by step guide to zeros of a polynomial

The zeros of a polynomial are the values of $$x$$ which satisfy the equation $$y = f(x)$$. Where $$f(x)$$ is a function of $$x$$, and the zeros of the polynomial are the values of $$x$$ for which the $$y$$ value is equal to zero. The number of zeros of a polynomial depends on the degree of the equation $$y = f (x)$$. All such domain values of the function whose range is equal to zero are called zeros of the polynomial.

Note: Graphically the zeros of the polynomial are the points where the graph of $$y = f(x)$$ cuts the $$x$$-axis.

### How to find zeros of polynomial?

The different types of equations and methods for finding their polynomial zeros are as follows:

Linear Equation:

A linear equation is of the form $$y=ax+b$$. Zero of this equation can be calculated by substituting $$y = 0$$, and on simplification we have $$ax+b=0$$ or $$x=-\frac{b}{a}$$.

There are two ways to factorize a quadratic equation:

1. The quadratic equation of the form $$x^2+x(a+b)+ab=0$$, can be factorized as $$(x + a)(x + b) = 0$$ and we have $$x = -a$$, and $$x = -b$$ as the zeros of the polynomial.
2. For a quadratic equation of the form $$ax^2+bx+c=0$$, which cannot be factorized, the zeros can be calculated using the formula method, and the formula is $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$.

Cubic Equation:

The cubic equation of the form $$ax^3+bx^2+cx+d$$, can be factorized by applying the remainder theorem. As per remainder theorem, we can substitute any smaller value for the variable $$x = α$$, and if the value of y reaches zero, $$y = 0$$, then $$(x – α)$$ is one root of the equation. In addition, we can divide the cubic equation by $$(x – α)$$ using the long division to obtain a quadratic equation. Finally, the quadratic equation can be solved by factorization or the formula method to obtain the two required roots of the equation.

Higher Degree Polynomial:

The higher degree polynomial equation is of the form $$y=ax^n+bx^n-1+cx^n-2+…px+q$$. These higher degree polynomials can be factorized using the remaining theorem to reach a quadratic equation. And the quadratic equation can be factorized to reach the two final factors required.

### Representing zeros of polynomial on graph

A polynomial expression in the form $$y = f (x)$$ can be represented on a graph across the coordinate axis. The value of $$x$$ is displayed on the $$x$$-axis and the value of $$f(x)$$ or the value of $$y$$ is displayed on the $$y$$-axis. A polynomial expression can be a linear expression, a quadratic expression, a cubic expression based on the degree of a polynomial. A linear expression represents a line, a quadratic equation represents a curve, and a higher degree polynomial represents a curve with uneven bends.

The zeros of a polynomial can be found from the graph by looking at the points where the graph line cuts the $$x$$-axis. The $$x$$ coordinates of the points where the graph cuts the $$x$$-axis are the zeros of the polynomial.

### Zeros of Polynomial – Example 1:

Find zeros of the polynomial function $$f(x)=x^3-12x^2+20x$$.

Solution:

First, take out $$x$$ as common:

$$f(x)=x(x^2-12x+20)$$

Now by splitting the middle term:

$$f(x)=x(x^2-2x-10x+20)$$

So we get:

$$f(x)=x[x(x-2)-10(x-2)]$$

$$f(x)=x(x-2)(x-10)$$

Here

$$x=0$$

$$x-2=0 → x=2$$

$$x-10=0 →x=10$$

Therefore, the zeros of polynomial function is $$x = 0$$ or $$x = 2$$ or $$x = 10$$.

## Exercises for Zeros of Polynomial

### Find the zeros of a polynomial.

1. $$\color{blue}{f(x)=3x^3-19x^2+33x-9}$$
2. $$\color{blue}{f(x)=x^2-10x+25}$$
3. $$\color{blue}{f(x)=x^3+2x^2-25x-50}$$
4. $$\color{blue}{f(x)=x^4+2x^{^3}-16x^2-32x}$$
1. $$\color{blue}{x=3 , x=\frac{1}{3}}$$
2. $$\color{blue}{x=5}$$
3. $$\color{blue}{x=-2, x=-5, x=5}$$
4. $$\color{blue}{x=0, x=-2, x=-4, x=4}$$

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