How to Find Zero of a Polynomial?

We can find the zeros of the polynomial for various types of polynomials using various methods that are discussed below.

• For Linear Polynomial
• For Quadratic Polynomial
• For Cubic Polynomial

For Linear Polynomial

For Linear Polynomials, finding zero is the easiest of all. as there is only one zero and that can also be calculated by simple rearrangement of the polynomial after the equating polynomial to 0.

For example, find zero for linear polynomial f(x) = 2x – 7.

Solution:

To find zero of f(x), equate f(x) to 0.

⇒ 2x – 7 = 0

⇒ 2x = 7 

⇒ x = 7/2

For Quadratic Polynomial

There are various methods to find roots or zeros of a quadratic polynomial such as splitting the middle term, a quadratic formula which is also known as the Shree Dharacharya formula, and completing the square which is somewhat similar to the quadratic formula, as quadratic formula comes from the completing the square for the general quadratic equation.

Learn more about solving quadratic equations or polynomials and how to solve them. The following examples show the method for finding zeros of Quadratic Polynomials in detail.

Example 1: Find out the zeros for P(x) = x² + 2x – 15. 

Answer:

x² + 2x – 15 = 0 

⇒ x² + 5x – 3x – 15 = 0

⇒ x(x + 5) –  3(x + 5) = 0 

⇒ (x – 3) (x + 5) = 0 

⇒ x = 3, -5

Example 2: Find the out zeros for P(x) = x² – 16x + 64. 

Answer:

x² – 16x + 64 = 0

Comparing with  ax² + bx + c = 0,

we get, a = 1, b = -16, and c = 64.

Thus, [Tex]x = \frac{-(-16) \pm \sqrt{(-16)^2 – 4(1)(64)}}{2(1)}[/Tex]

[Tex]\Rightarrow x = \frac{16 \pm \sqrt{ 256- 256}}{2}[/Tex]

[Tex]\Rightarrow x = \frac{16 \pm 0}{2}[/Tex]

⇒  x = 8, 8

For Cubic Polynomial

To find zeros of cubic there are many ways, such as rational root theorem and long division together. One method of finding roots of cubic or any higher degree polynomial is as follows:

Step 1: Use the rational root theorem to find the possible roots. i.e., If a polynomial has a rational root it must be the division of p/q, where p is the integer constant and q is the leading coefficient.

Step 2: After finding one root, divide the polynomial with the factor formed by that root using long division and write the polynomial as a product of quotient and dividend.

Step 3: If the quotient is a quadratic expression solve it by the methods above mentioned for quadratic polynomials. If not a polynomial of a degree  2 then repeat steps 1 and 2 until the quotient becomes a polynomial with degree 2.

Step 4: The result of step 3 is the required factors, and by equating the factor to 0, we can find the zeros of the polynomial.

Example: Find the zeros of the cubic polynomial p(x) = x³ + 2x² – 5x – 6.

Solution:

p(x) = x³ + 2x² – 5x – 6

As p/q = -6

By rational root theorem, all possible rational roots of the polunomial are divisors of p/q. 

Thus, divisors = ±1, ±2, ±3, ±6

x = -1, in p(x), we get

p(-1) = (-1)³ + 2(-1)² – 5(-1) – 6

⇒ p(-1) = -1 + 2 + 5 – 6 = 0

Thus, by factor theorem, x + 1 is the factor of p(x).

Thus, x³ + 2x² – 5x – 6 = (x+1)(x² +x – 6)

⇒ x³ + 2x² – 5x – 6 = (x+1)(x-2)(x+3)

For zeroes, p(x) = 0,

Zeros of p(x) are x = -1, x = 2, and  x = -3.

Zeros of a Polynomial are those real, imaginary, or complex values when put in the polynomial instead of a variable, the result becomes zero (as the name suggests zero as well). Polynomials are used to model some physical phenomena happening in real life, they are very useful in describing situations mathematically.

The zeros of a polynomial are all the x-values that make the polynomial equal to zero. Zeroes of a polynomial tell us about the x-intercepts of the polynomial’s graph. In this article, we will discuss about the zeroes of a polynomial, how to find them, the factor theorem, etc.