# Which of the following is a zero of the function f(x) = 3 - 7x^2 + 7x + 15.

0 votes
Which of the following is a zero of the function f(x) = x3 - 7x2 + 7x + 15.
 A. -5 B. -3 C. -1 D. 1

asked Sep 3, 2014

## 1 Answer

0 votes

Best answer

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation anx n + an  1x n – 1 + ... + a1x + a0 = 0, then p is a factor of a0 and q is a factor if an.

The function is x 3 - 7x 2 + 7x + 15 = 0.

If p/q is a rational zero, then p is a factor of 15 and q is a factor of 1.

The possible values of p are   ± 1, ± 3, ± 5 and ± 15.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p / q = ± 1, ± 3,  ± 5 and ± 15.

Make a table for the synthetic division and test possible real zeros.

 p/q 1 - 7 7 15 1 1 - 6 1 16 - 1 1 - 8 15 0

Since, f(- 1) = 0, x = - 1 is a zero. The depressed polynomial is x 2 - 8x + 15 = 0.

By factor by grouping.

x 2 - 8x + 15 = 0

x 2 - 3x - 5x + 15 = 0

x(x - 3) - 5(x - 3) = 0

Factor : (x - 5)(x - 3) = 0

Apply zero product property.

x - 5 ⇒ x = 5.

x - 3 ⇒ x = 3.

Therefore, the roots of the function are x = - 1, x = 5, and x = 3.

The option C is correct.

answered Sep 3, 2014
selected Sep 3, 2014 by tonymate