SAT Level 5 PAM Problem with Solution

SAT Level 5 Passport to Advanced Math Problem
with Solution

Today I would like to provide a solution for the difficult Passport to Advanced Math SAT problem I recently posted. Here is the original post: SAT Level 5 PAM Problem

Level 5 Passport to Advanced Math

g(x)=x² + 4x – 1
h(x)=2x³ + 3x² + x

The polynomials g and h are defined above. Which of the following polynomials is divisible by 2x – 1 ?

A) k(x) = g(x) – h(x)
B) k(x) = 12g(x) – h(x)
C) k(x) = g(x) – 10h(x)
D) k(x) = 12g(x) – 10h(x)

First recall that the factor theorem says that p(r) = 0 if and only if x – r is a factor of the polynomial p.

* Solution using the factor theorem: We note that 2x – 1 = 2(x – 1/2), and use r = 1/2. We have

g(1/2) = (1/2)² + 4(1/2) – 1 = 1/4 + 2 – 1 = 1/4 + 1 = 5/4

h(1/2) = 2(1/2)³ + 3(1/2)² + 1/2 = 2(1/8) + 3(1/4) + 1/2 = 1/4 + 3/4 + 1/2 = 3/2

Since 12(5/4) – 10(3/2) = 15 – 15 = 0, we see that if k(x) = 12g(x) – 10h(x), then k(1/2) = 0, and it follows that 2x – 1 is a factor of k. So, the answer is choice D.

Note: To use the factor theorem, we need to divide by a linear polynomial of the form x – r. So, we rewrite 2x – 1 as 2(x – 1/2). We see that 2x – 1 is a factor of the polynomial if and only if x – 1/2 is a factor of the polynomial. So, we use r = 1/2.

Share this problem with your Facebook friends that are preparing for the SAT