Multiple choice

𝑥³﹣𝑝𝑥² + 63𝑥﹣90 = 0 is a polynomial equation, all of whose roots are positive integers. Which of the following gives the value of 𝑝?

  1. 12

  2. 13

  3. 14

  4. 15
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let roots be r1, r2, r3. r1*r2*r3 = 90. r1+r2+r3 = p. r1r2 + r2r3 + r3r1 = 63. Possible factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15... Testing sets: 2, 5, 9 (sum 16, prod 90, sum of products 10+45+18=73 no). Testing 3, 5, 6 (sum 14, prod 90, sum of products 15+30+18=63 yes). So p = 14.

AI explanation

By Vieta's formulas for x^3 - px^2 + 63x - 90 = 0, the product of the three positive integer roots equals 90, and the sum of their pairwise products equals 63. The only positive integers satisfying these conditions are 3, 5, and 6, since their pairwise products give 15 + 18 + 30 = 63. The value of p is the sum of these roots: 3 + 5 + 6 = 14.

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