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

Question

Which of the following is a zero of the function f(x) = x³ - 7x² + 7x + 15.

	A.	-5
	B.	-3
	C.	-1
	D.	1

 

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 - 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.