College Math Placement Practice Test

If a polynomial P(x) has roots at x = 2 and x = -3, what is one possible factorization of P(x)?

1. P(x) = k(x - 2)(x + 3)

2. P(x) = k(x + 2)(x - 3)

3. P(x) = k(x - 2)(x - 3)

4. P(x) = k(x + 2)(x + 3)

The correct answer is: P(x) = k(x - 2)(x + 3)

To understand why the first choice is a valid factorization of the polynomial P(x), let's recall how the roots of a polynomial relate to its factors. If a polynomial has roots at x = 2 and x = -3, this means that the polynomial will equal zero at these points. For the root x = 2, the corresponding factor is (x - 2), because when you substitute 2 into the polynomial, it should yield zero: P(2) = k(2 - 2)(2 + 3) = 0. For the root x = -3, the corresponding factor is (x + 3), which similarly satisfies the condition that when we substitute -3, the polynomial equals zero: P(-3) = k(-3 - 2)(-3 + 3) = 0. Thus, by multiplying these factors together and including a non-zero constant k (which can be any real number to stretch or compress the polynomial vertically without altering the roots), we get P(x) = k(x - 2)(x + 3). The other answer choices do not reflect the correct roots based on the required factors. Therefore, the factorization in the first choice accurately describes