# How to Find Zeros of Polynomials?

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 the values of $$x$$ for which the polynomial equals zero. In other words, they are the solutions of the equation formed by setting the polynomial equal to zero. The zeros of a polynomial can be real or complex numbers, and they play an essential role in understanding the behavior and properties of the polynomial function.

## A step-by-step guide to finding 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.

Finding the zeros (roots) of a polynomial can be done through several methods, including:

1. Factoring: Find the polynomial factors and set each factor equal to zero.
2. Synthetic Division: Divide the polynomial by a linear factor $$(x – c)$$ to find a root c and repeat until the degree is reduced to zero.
3. Graphical Method: Plot the polynomial function and find the $$x$$-intercepts, which are the zeros.
4. Newtons Method: An iterative method to approximate the zeros using an initial guess and derivative information.
5. Bairstow Method: A complex extension of the Newtons Method for finding complex roots of a polynomial.

The method used will depend on the degree of the polynomial and the desired level of accuracy.

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 polynomials?

There are several types of equations and methods for finding their polynomial zeros:

1. Linear Equations (Degree 1 Polynomial): Zeros can be found by solving for $$x$$ using the formula $$x =-\frac{b}{a}$$, where $$a$$ and $$b$$ are coefficients.
2. Quadratic Equations (Degree 2 Polynomials): Zeros can be found using the Quadratic Formula $$x=\frac{\left(-b\pm \:\:\left(\sqrt{b^2-4ac}\right)\right)}{2a}$$, where $$a, b,$$ and $$c$$ are coefficients.
3. Cubic Equations (Degree 3 Polynomials): Zeros can be found using either the Rational Root Theorem or the Synthetic Division.
4. Higher Degree Polynomials (Degree 4 or higher): Zeros can be found using the Rational Root Theorem, Synthetic Division, Newton-Raphson Method, or the Bairstow Method.
5. Complex Polynomials: Zeros can be found using the Complex Conjugate Root Theorem, or by graphing the polynomial in the complex plane.

Note: The choice of method depends on the complexity of the polynomial and the desired level of accuracy.

### Representing zeros of polynomials 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, quadratic, or 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 in 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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