In the “Proving Grounds” Quiz 1 #5, this question is given:

p(x) = ax^2 + bx + c
q(x) = ax^2 + bx + d

The functions p(x) and q(x) are defined above, and a, b, c, and d are constants. If x – 3 is a factor of p(x) and d is 8 greater than c, what is the remainder when q(x) is divided by x – 3?

In your answer explanation, you state:
“…We know that, as a general rule, the remainder when a polynomial f(x) is divided by x – a is f(a).”

Can you explain this further?

This is an application of the polynomial remainder theorem; I think there’s a really nice and thorough explanation of it at Purple Math, so I’ll refer you there.

This concept appears in various forms throughout the released practice tests from College Board, although often not in such an abstract way. The closest example to my Proving Grounds question is probably Test 1 Section 4 #29.

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