Question

Which of the following is a zero of the function f(x) = 3x³ + 12x² + 3x - 18.

	A.	-1
	B.	1
	C.	2
	D.	3

 

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 3x ³ + 12x ² + 3x - 18 = 0.

If p/q is a rational zero, then p is a factor of - 18 and q is a factor of 3.

The possible values of p are   ± 1, ± 2, ± 3, ± 6 and ± 18.

The possible values for q are ± 1 and ± 3.

By the Rational Roots Theorem, the only possible rational roots are, p / q = ± 1, ± 2, ± 3, ± 6, ± 18,  ± 1/3, and ± 2/3.

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

p/q	3	12	3	- 18
1	3	15	18	0

Since, f(1) = 0, x = 1 is a zero. The depressed polynomial is 3x ² + 15x + 18 = 0.

By factor by grouping.

Divide each side by 3.

x ² + 5x + 6 = 0

x ² + 2x + 3x + 6 = 0

x(x + 2) + 3(x + 2) = 0

Factor : (x + 3)(x + 2) = 0

Apply zero product property.

x + 3 ⇒ x = - 3.

x + 2 ⇒ x = - 2.

Therefore, the roots of the function are x = 1, x = - 3, and x = - 2.

The option B is correct.