Question

If f(x)=x³+2x²-41x-42 , which of the following is a factor of f(x) ?

a. (x-7)
b. (x-1)
c. (x+9)
d. (x+1)

Answer 1

Using the Remainder Theorem, it is found that (x - 7) is a factor of the polynomial f(x) = x³ + 2x² - 41x - 42 because when x = 7, the polynomial equals zero.

Step-by-step explanation:

To determine which of the given options is a factor of the polynomial f(x) = x³ + 2x² - 41x - 42, we can use the Remainder Theorem. According to this theorem, if a polynomial f(x) has a factor of (x - r), then the polynomial will equal zero when x = r. We can test each option by substituting the value of x from the factor into the polynomial expression.

• (x - 7): When we substitute x = 7 into the equation, we get 7³ + 2 · 7² - 41 · 7 - 42 = 0, which tells us that (x - 7) is indeed a factor of f(x).
• (x - 1): Substituting x = 1 gives us 1³ + 2 · 1² - 41 · 1 - 42, which does not equal zero, so (x - 1) is not a factor.
• (x + 9): Substituting x = -9 gives us (-9)³ + 2 · (-9)² - 41 · (-9) - 42, which does not equal zero, so (x + 9) is not a factor.
• (x + 1): Substituting x = -1 gives us (-1)³ + 2 · (-1)² - 41 · (-1) - 42, which does not equal zero, so (x + 1) is not a factor.

Therefore, the correct answer is (x - 7), which means option a is the factor of f(x).