2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 7 Easy

For how many values of the constant k will the polynomial x^{2}+kx+36 have two distinct integer roots?

  • A.

    6

  • B.

    8

  • C.

    9

  • D.

    14

  • E.

    16

Answer:B

By Vieta's Formulae, the product of two roots is 36=2^2\times 3^2, which contains 9 factors. Break down 36 to get 36=1\times 36=2\times 18=3\times 12=4\times 9=6\times 6, and the signs can be reversed. Since we want distinct roots, the 6\times 6 case will be neglected, so there are 4\times 2=8 choices of k.

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