Difference between revisions of "2024 AMC 12A Problems/Problem 15"
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The roots of x^3 + 2x^2 − x + 3 are p, q, and r. What is the value of (p^2 + 4)(q^2 + 4)(r^2+4)? | The roots of x^3 + 2x^2 − x + 3 are p, q, and r. What is the value of (p^2 + 4)(q^2 + 4)(r^2+4)? | ||
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| + | ==Solution== | ||
| + | You can factor (p^2 + 4)(q^2 + 4)(r^2 + 4) as (p − 2i)(p + 2i)(q − 2i)(q + 2i)(r − 2i)(r + 2i). | ||
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| + | For any polynomial f(x), you can create a new polynomial f(x+2), which will have roots that instead have the value subtracted. | ||
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| + | Substituting x-2 and x+2 into x for the first polynomial, gives you 10i-5 and -10i-5 as c for both equations. Multiplying 10i-5 and -10i-5 together gives you **125** | ||
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| + | -ev2028 | ||
Revision as of 17:56, 8 November 2024
The roots of x^3 + 2x^2 − x + 3 are p, q, and r. What is the value of (p^2 + 4)(q^2 + 4)(r^2+4)?
Solution
You can factor (p^2 + 4)(q^2 + 4)(r^2 + 4) as (p − 2i)(p + 2i)(q − 2i)(q + 2i)(r − 2i)(r + 2i).
For any polynomial f(x), you can create a new polynomial f(x+2), which will have roots that instead have the value subtracted.
Substituting x-2 and x+2 into x for the first polynomial, gives you 10i-5 and -10i-5 as c for both equations. Multiplying 10i-5 and -10i-5 together gives you **125**
-ev2028
