SAT Math: The Remainder Theorem

The Remainder Theorem is not something that you will use many times when taking the SAT, but it has shown up on a couple of problems in the practice tests that have been released in The Official SAT Study Guide.

The Remainder Theorem says that if you divide some polynomial p(x) by the linear factor x – a, the remainder that you get is equal to p(a). For example, consider the problem below where the polynomial p(x) = x² – 2x – 11 is divided by x – 5 using either Long Division or Synthetic Division.

The remainder in both cases is 4. The Remainder Theorem says that if the remainder when you divide by x – 5 is 4, then it is also true that p(5) = 4.  It is easy to check that this is indeed true.

p(x) = x² – 2x – 11
p(5) = 5² – 2(5) – 11
p(5) = 25 – 10 – 11
p(5) = 4

There is also a relationship between the polynomial p(x), the divisor (– a), the quotient q(x) and the remainder r. Specifically

p(x) = q(x)⋅(x – a) + r

Using the example above, you can write

p(x) = (x + 3)(x – 5) + 4
p(x) = x² – 5x + 3x – 15 + 4
p(x) = x² – 2x – 11

OK, let’s try a few problems. When you’re done, scroll down for the answers.

1. A student used division to divide a polynomial p(x) by x – 3 and got a remainder of 5. Which of the following statements about p(x) must be true?

A) x – 3 is a factor of p(x).
B) x – 5 is a factor of p(x)
C) The value of p(-3) is 5.
D) The value of p(3) is 5.

2. When a polynomial p(x) is divided by x – 4, the quotient is another polynomial q(x), and the remainder is -6. Which of the following must be true of p(x)?

A) p(x) = (x – 4)(x – 6)
B) p(x) = (x – 4)(x + 6)
C) p(x) = q(x)(x – 4) – 6
D) p(x) = q(x)·(x – 4) + 6

3. When a polynomial p(x) is divided by x + 3, the quotient q(x) is x – 8 and the remainder r is 5. Which of the following is the polynomial p(x)?

A) p(x) = x2 – 5x – 29
B) p(x) = x2 – 5x – 19
C) p(x) = x2 + 5x + 19
D) p(x) = x2 + 5x + 29


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(Scroll down for answers)
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1. D
2. C
3. B
p(x) = (x + 3)(x – 8) + 5
p(x) = x² – 8x + 3x – 24 + 5
p(x) = x² – 5x – 19

If you have questions about these problems or anything else related to the SAT, send us an email at info@cardinalec. com.

 

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