Question

For homework yesterday, Jordan divided the polynomial P(x) by (x² - 4). Today his work is smudged, and he cannot read P(x). The only parts of his work he can read are the quotient (x-1) and the remainder (x + 4). His teacher has asked him to find P(-2). What is P(-2)?

a. -12
b. -10
c. -8
d. -6

Answer 1

Jordan's polynomial P(-2) equals 2, which is found by evaluating the remainder at x = -2, and recognizing that x = -2 is actually a root of the polynomial due to the factor (x + 2) of the divisor (x² - 4).

Step-by-step explanation:

To find P(-2), we can apply the remainder theorem which tells us that when a polynomial P(x) is divided by (x - a), the remainder is P(a). Since Jordan's polynomial P(x) was divided by (x² - 4), and the remainder was (x + 4), we can find P(-2) by simply evaluating the remainder at x = -2.

The remainder polynomial is (x + 4). Substituting x = -2 into this gives us:

P(-2) = (-2) + 4 = 2.

However, because the divisor is (x² - 4), which is the difference of squares and can be factored into (x + 2) and (x - 2), we realize that the division must have been by (x + 2) which means x = -2 is actually a zero of the polynomial. Hence, P(-2) is not the remainder but instead the quotient times the divisor plus the remainder evaluated at x = -2.

The quotient given is (x - 1), and the division factor relevant to x = -2 is (x + 2). So we calculate:

(-2 - 1)(-2 + 2) + 2 = -3(0) + 2 = 2

Therefore, P(-2) equals 2.