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2024 AMC 12A Problem 15

Problem 15 of 25IntermediateAlgebra

The roots of x3+2x2−x+3x^3+2x^2-x+3 are pp, qq, and rr. What is the value of (p2+4)(q2+4)(r2+4)? (p^2+4)(q^2+4)(r^2+4)?

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Solution

Since P(x)=(x−p)(x−q)(x−r),P(x)=(x-p)(x-q)(x-r), grouping p2+4=(p−2i)(p+2i)p^2+4=(p-2i)(p+2i) over all roots gives ∏(p2+4)=P(2i) P(−2i). \prod(p^2+4)=P(2i)\,P(-2i). Compute P(2i)=−8i−8−2i+3P(2i)=-8i-8-2i+3 =−5−10i=-5-10i and P(−2i)=8i−8+2i+3P(-2i)=8i-8+2i+3 =−5+10i.=-5+10i. Their product is (−5)2+102=25+100=125.(-5)^2+10^2=25+100=125. Thus, the correct answer is D.

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Concepts: complex number · polynomial

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