WAEC 2016 · Paper 2 · Q4

(a)
When $f(x) = 2x^3 + mx^2 + nx + 11$ is divided by $x^2 + 5x + 1$ , the quotient is $2x - 5$ and the remainder is $30x + 16$ . Find the values of $m$ and $n$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

The polynomial is the divisor times the quotient, plus the remainder: $f(x) = (x^2 + 5x + 1)(2x - 5) + (30x + 16)$ .
Multiply out term by term: $x^2(2x - 5) = 2x^3 - 5x^2$ , $5x(2x - 5) = 10x^2 - 25x$ and $1(2x - 5) = 2x - 5$ .
Add them: $2x^3 + 5x^2 - 23x - 5$ .
Add the remainder: $2x^3 + 5x^2 - 23x - 5 + 30x + 16 = 2x^3 + 5x^2 + 7x + 11$ .
Compare with $2x^3 + mx^2 + nx + 11$ : $m = 5$ and $n = 7$ .
The constants, 11, agree.