Question

Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

Answer 1

A. x = 3, -1, -5

f(x) = (x - 3)(x + 1)(x + 5)

Step-by-step explanation

Given polynomial: f(x) = x³ + 3x² - 13x - 15

Putting x = 3 into f(x), we have

f(3) = 3³ + 3(3)² - 13(3) - 15 = 0

That is, x - 3 is factor of f(x)

To know other factors, we factorize f(x) using long division method

(x - 3)(x² + 6x + 5)

(x - 3)(x² + 5x + x + 5)

(x - 3)(x + 1)(x + 5)

x = 3, -1, -5

To use the real zeros to factor f, we have

f(x) = (x - 3)(x + 1)(x + 5)