Question

Is the binomial a factor of the polynomial function? f(x) = x³ + 3x² − 25x − 75 Select Yes or No for each binomial. (x - 1) (x - 3) (x + 3) (x - 5) (x + 5)

Answer 1

Using the remainder theorem to test each binomial, (x + 3), (x - 5), and (x + 5) are factors of the polynomial f(x) = x³ + 3x² − 25x − 75, because the remainder is zero when the polynomial is evaluated at the roots of these binomials.

Step-by-step explanation:

To determine if a binomial is a factor of a polynomial, one can use either synthetic division or the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by a binomial (x - c) and the remainder is 0, then (x - c) is a factor of f(x).

Let's apply the remainder theorem to the given polynomial f(x) = x³ + 3x² − 25x − 75 and the given binomials.

For (x - 1): f(1) = 1³ + 3(1)² - 25(1) - 75 = 1 + 3 - 25 - 75 = -96. So, No.

For (x - 3): f(3) = 3³ + 3(3)² - 25(3) - 75 = 27 + 27 - 75 - 75 = -96. So, No.

For (x + 3): f(-3) = (-3)³ + 3(-3)² - 25(-3) - 75 = -27 + 27 + 75 - 75 = 0. So, Yes.

For (x - 5): f(5) = 5³ + 3(5)² - 25(5) - 75 = 125 + 75 - 125 - 75 = 0. So, Yes.

For (x + 5): f(-5) = (-5)³ + 3(-5)² - 25(-5) - 75 = -125 + 75 + 125 - 75 = 0. So, No, there is an arithmetic mistake here, let's recalculate: (-5)³ + 3(-5)² - 25(-5) - 75 = -125 + 75 + 125 - 75 = 0. Correct result is Yes.

Therefore, the factors of the polynomial function f(x) are (x + 3), (x - 5), and (x + 5).