ZEROS OF A POLYNOMIAL

Published on: March 28, 2024

Zeroes of a Polynomial

To find the value of a polynomial at x = a, substitute x with a and do the calculation. For example, consider the polynomial eq. To find the value of p(x) at x = 3, replace x with the value 3 and we get

p(3) = 5 × 32 - 2 × 3 + 3        =  45 - 6 + 3        = 42

A real number a is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called the root of the equation p(x) = 0.

Consider the polynomial x2 - 5x + 6. Replace x by 2, then we get

22 - 5 × 2 + 6 = 4 - 10 + 6 = 0

∴  we say 2 is a zero of the polynomial or 2 is the root of the polynomial x2 - 5x + 6

By checking, we can see that 3 is also a zero of the polynomial.

Note

  • A zero of a polynomial can be any real number.

  • 0 need not be the zero of a polynomial.

  • Every linear polynomial has one and only one zero or root.

  • Every quadratic polynomial has two zeros.

  • Every cubic polynomial has three zeros.

  • A polynomial can have one or more zeros.

  • A non-zero constant polynomial has no zero.

Example 1

Check whether −1 is a zero of the polynomial x3 + 7x - 1

Solution

Replace x by −1.

(-13) + 7 × -1 - 1 = -1 - 7 -1 ≠ 0

∴ −1 is not a zero of the polynomial.

Example 2

Find a zero of the polynomial x +1.

Solution

To find the zero of the polynomial, equate x +1 to 0, i.e., x +1 = 0 ⟹ x = −1.

∴ −1 is a zero of the polynomial

Example 3

Verify whether 2 and 0 are zeroes of the polynomial a2 - 2a

Solution

p(a) = a2 -2ap(0) = 02 - 2 × 0 = 0p(2) = 22 - 2 × 2 = 0

Hence, 2 and 0 are the zeroes of the polynomial.