Question

Which of the following is a zero of the function f(x) = 2x³ + 3x² - 3x - 2.

	A.	-2
	B.	-1
	C.	1212121
	D.	2

c1/3

 

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 2x ³ + 3x ²- 3x - 2 = 0.

If p/q is a rational zero, then p is a factor of - 2 and q is a factor of 2.

The possible values of p are   ± 1 and ± 2.

The possible values for q are ± 1 and ± 2.

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

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

p/q	2	3	- 3	- 2
1	2	5	2	0

Since, f(1) = 0, x = 1 is a zero. The depressed polynomial is 2x ² + 5x + 2 = 0.

By factor by grouping.

2x ² + 5x + 2 = 0

2x ² + 4x + x + 2 = 0

2x(x + 2) + 1(x + 2) = 0

Factor : (2x + 1)(x + 2) = 0

Apply zero product property.

2x + 1 = 0 and x + 2 =  0

⇒ x = - 1/2 and x = - 2.

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

The option A is correct.