For what value of k does the rational expression (x^2 - kx + 9)/(x - 3) simplify to a linear polynomial with no remainder? Find the value of k.

For what value of $k$ does the rational expression $\frac{x ^{2} - k x + 9}{x - 3}$ simplify to a linear polynomial with no remainder? Find the value of $k$ .

hard 3 marks Class 7 · Mathematics · Algebraic Expressions

Answer

The correct answer is 6 — Solve for $k$ : $3 k = 18 \to k = 6$ .

For the fraction to have no remainder when divided by $(x - 3)$ , the numerator must be divisible by $(x - 3)$ .
Use the Remainder Theorem: evaluate the numerator at $x = 3$ . Set $3^{2} - k (3) + 9 = 0$ .
Simplify: $9 - 3 k + 9 = 0 \to 18 - 3 k = 0$ .
Solve for $k$ : $3 k = 18 \to k = 6$ .

Common mistake: Solving for k by incorrect sign or ignoring that remainder must be zero

Concept tested: Remainder theorem for rational expressions