Which of the following is an actual zero of f(x)=x^3+4x^2-4x-16?

Question

A. -16 
B. -4 
C. 0 
D. 4

Answer 1

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 a_nx ⁿ + a_n – 1x ^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

The function is x ³ + 4x ² - 4x - 16 = 0.

If p/q is a rational zero, then p is a factor of - 16 and q is a factor of 1.

The possible values of p are ± 1, ± 2, ± 4, ± 8, and ± 16.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 4, ± 8, and ± 16.

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

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

p/q	1	4	- 4	- 16
1	1	5	1	- 15
- 1	1	4	- 8	- 8
2	1	6	8	0

Since, f(2) = 0, x = 2 is a zero. The depressed polynomial is  x ² + 6x + 8 = 0.

 

Answer 2

continued ---->

Since the depressed polynomial of this zero, f(x) = x ² + 6x + 8, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

x = [ - b ± √(b ² - 4ac ) ] / 2a.

Substitute a = 1, b = 6, and c = 8.

x = [ - (6) ± √((6)² - 4 * 1 * 8) ] / 2 * 1

x = [ - 6 ± √(36 - 32) ] / 2

x = [ - 6 ± 2 ] / 2

x = - 3 ± 1.

x = - 3 + 1 = - 2 and x = - 3 - 1 = - 4.

Therefore, the roots of the function are 2, - 2 and - 4.

The option B is correct answer.