Polynomial Problems

Algebra Level 2

The polynomial $x^{3} + px^{2} - x + q$ has a factor $(x - 5)$ and a remainder of 24 when divided by $(x - 1)$ .

Find the values of $p$ and $q$ .

5 and 1 -5 and -1 6 and -30 -6 and 30

1 solution

Ahmad Khamis
Aug 2, 2015

let $f(x) = x^{3}-px^{2}-x+q$ . Since $x-5$ is a factor of $f(x)$ , $f(5) = 0$ . Thus, $125 + 25p -5 +q =0$ which simplifies to $25p + q = -120$ . Also, $f(1) = 24$ because if a polynomial is divided by $(x-a)$ , the remainder is $f(a)$ . Thus, $1 + p -1 + q = 24$ which simplifies to $p+q = 24$ . Solving both equations simultaneously yields $q = 30$ and $p = -6$

Moderator note:

Simple standard approach of applying the Remainder-Factor theorem.

Polynomial Problems

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