How to Find Zeros of Polynomials?

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.

How to Represent zeros of polynomials on the 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. \(\color{blue}{x=0, x=-2, x=-4, x=4}\)