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

The all possible rational zeroes of f(x) are
`\pm 1,\pm (1)/(2),\pm (1)/(4),\pm7,\pm (7)/(2),\pm (7)/(4),`

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 (1,\:7)/(1,\:2,\:4)=\pm 1,\pm (1)/(2),\pm (1)/(4),\pm7,\pm (7)/(2),\pm (7)/(4),`

Hence, the all possible rational zeroes of f(x) are
`\pm (1,\:7)/(1,\:2,\:4)=\pm 1,\pm (1)/(2),\pm (1)/(4),\pm7,\pm (7)/(2),\pm (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.