WAEC 2017 · Paper 2 · Q4

If $(x + 1)$ and $(x - 2)$ are factors of the polynomial $g(x) = x^4 + ax^3 + bx^2 - 16x - 12$ , find the values of $a$ and $b$ .

Move a and b until the curve passes through both marked points. Which values work?

Worked solution (try it first)

$(x + 1)$ is a factor, so $g(-1) = 0$ : $1 - a + b + 16 - 12 = 0$ , which gives $a - b = 5$ .
$(x - 2)$ is a factor, so $g(2) = 0$ : $16 + 8a + 4b - 32 - 12 = 0$ , which gives $8a + 4b = 28$ .
Divide the second equation by 4: $2a + b = 7$ .
Add it to $a - b = 5$ : $3a = 12$ , so $a = 4$ .
Then $b = a - 5 = -1$ .
Check: $g(x) = x^4 + 4x^3 - x^2 - 16x - 12$
$= (x + 1)(x - 2)(x + 2)(x + 3)$ ✓.