Question

Which function, when divided by x + 2, yields a remainder of 3?

A. f(x) = x² + x + 5

B. f(x) = x² + 3x + 5

C. f(x) = x² - 4x + 7

D. f(x) = x² - 6x + 11

Answer 1

Using the remainder theorem, the function that yields a remainder of 3 when divided by x + 2 is f(x) = x² + 3x + 5, which corresponds to option B.

Step-by-step explanation:

The student asked which function, when divided by x + 2, yields a remainder of 3. To determine this, we can use the remainder theorem, which states that if a polynomial f(x) is divided by x - k, then the remainder is f(k). In this case, for division by x + 2, the value of k is -2.

Let's evaluate each option at x = -2:

• f(x) = x² + x + 5: f(-2) = (-2)² - 2 + 5 = 4 - 2 + 5 = 7
• f(x) = x² + 3x + 5: f(-2) = (-2)² + 3(-2) + 5 = 4 - 6 + 5 = 3
• f(x) = x² - 4x + 7: f(-2) = (-2)² - 4(-2) + 7 = 4 + 8 + 7 = 19
• f(x) = x² - 6x + 11: f(-2) = (-2)² - 6(-2) + 11 = 4 + 12 + 11 = 27

As we can see, the function that yields a remainder of 3 when x = -2 is f(x) = x² + 3x + 5, which is option B.