Question

Consider the cubic function y = f(x) = -4x^3 – 6x^2 + 3x + 7.

Use the Rational Zeroes Theorem to list the possible rational zeroes of f(x). ____________

Answer 1

Answer:

The all possible rational zeroes of f(x) are $\pm 1,\pm \frac{1}{2},\pm \frac{1}{4},\pm7,\pm \frac{7}{2},\pm \frac{7}{4},$

Step-by-step explanation:

Consider the provided cubic function.

$y = f(x) = -4x^3 -6x^2 + 3x + 7$

Rational zeros Theorem states that:

If P(x) is a polynomial and if p/q is a zero of P(x), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).

Therefore,

The factor of constant term: ±1, ±7

Factors of leading coefficient: ±1, ±2, ±4

The Possible values of p/q: $\pm \frac{1,\:7}{1,\:2,\:4}=\pm 1,\pm \frac{1}{2},\pm \frac{1}{4},\pm7,\pm \frac{7}{2},\pm \frac{7}{4},$

Hence, the all possible rational zeroes of f(x) are $\pm \frac{1,\:7}{1,\:2,\:4}=\pm 1,\pm \frac{1}{2},\pm \frac{1}{4},\pm7,\pm \frac{7}{2},\pm \frac{7}{4},$

Note: These are all possible rational zeroes.

The rational zeros of the provided cubic function is x=1.

Substitute x=1 in provided function.

$-4\cdot \:1^3-6\cdot \:1^2+3\cdot \:1+7=0\\0=0$

Which is true.

Hence, the rational zeros of the provided cubic function is x=1.