Real Zeros of a Polynomial

The real zeros of a polynomial are the x-intercepts of the polynomial.

(The following observations are consequences of the material in the section on the leading coefficient test.)

Observation 1

Every polynomial of odd degree must have at least one real zero . This is because when the degree is odd the range is all real numbers and consequently the graph must cross the x-axis.

Observation 2

A polynomial of even degree may or may not have real zeros . This is because the range of such a polynomial is limited and may or may not include zero. When a polynomial with even degree has real zeros, it has an even number of them .

Example 1

Find all real zeros to

Solution .

We must solve for x. Factoring further we get . Thus, x = 0, x = 5, and x = -5. The graph below confirms these solutions.

Example 2

Find the real zeros to

Solution .

By factoring we have . This gives x = 0 as the only real zero (and only x-intercept) since has no real solutions.

Example 3

Find the real zeros of

Solution .

Setting this equal to zero and factoring out -1, we have . Factoring further gives us . The only real solutions are x = 3 and x = -3.

Example 4

Find the zeros of

Solution .

Factoring we get . Since the two factors do not have real solutions, this polynomial has no real zeros. Hence the graph does not have x-intercepts.

Example 5

Find the real zeros for

Solution .

Using grouping we have, . This gives us which yields . The first two factors give us the multiple root of x = 1. Since the equation has no real roots, the only real zero is x =1 which has multiplicity of two.