PSAT Math Multiple-Choice Question 698: Answer and Explanation

Question: 698

• A. 2i - 3
• B. 3i - 1
• C. 3 - i
• D. 2 + 3i

Correct Answer: B

Explanation:

(B) For this problem, we can employ a little trick so that we have an i² term and can factor out an i in the numerator. We can use the knowledge that i² = -1 to change -6 to 6(-1) or 6i². Thus, we can rewrite the original numerator as:

i³ + 6i² + 9i

An i can be factored out, leaving us with the following in the numerator:

i(i² + 6i + 9)

The part inside the parentheses can be factored as:

(i + 3)²

Therefore, our whole expression can be rewritten as:

Since i + 3 is the same as 3 + i, we can cancel one of the i + 3 terms from the numerator with the 3 + i term in the denominator. This leaves:

i(i + 3) or i² + 3i

Because we know that i² = -1, we can rewrite this as -1 + 3i, which is the same as choice (B).

Alternatively, we can first simplify the numerator since i² = -i:

i³ + 9i - 6 = -i + 9i - 6 = 8i - 6

Now notice that none of the answer choices are fractions, so we want to get rid of the denominator. We can get rid of the i-term in the denominator by multiplying by its complex conjugate. This is found by changing the sign of all terms with an i in them and leaving the signs of all real numbers the same. So in our case, the complex conjugate of the denominator 3 + i is 3 - i. We multiply both the numerator and denominator by this conjugate:

By using FOIL and simplifying, we have:

Factoring out a 10 from the numerator gives:

Choice (B) is the answer.