For which value of r does (x^2 + r x + 9)/(x + 3) reduce to a linear polynomial with no remainder?

For which value of $r$ does $\frac{x ^{2} + r x + 9}{x + 3}$ reduce to a linear polynomial with no remainder?

hard 3 marks Class 7 · Mathematics · Algebraic Expressions

Answer

The correct answer is 6 — Solve for $r$ : $3 r = 18; R i g h t a r r o w; r = 6$ .

Use the remainder theorem: a polynomial $P (x)$ divided by $(x - a)$ has remainder $P (a)$ .
Here the divisor is $(x + 3) = (x - (- 3))$ , so set $x = - 3$ .
Evaluate the numerator at $x = - 3$ : $(- 3)^{2} + r (- 3) + 9$ .
Simplify: $9 - 3 r + 9 = 18 - 3 r$ .
For no remainder, set $18 - 3 r = 0$ .
Solve for $r$ : $3 r = 18; R i g h t a r r o w; r = 6$ .

Common mistake: Using

x = 3

instead of

x = - 3

Concept tested: Rational expression simplification