Divide x^2 - 12x + 5 by 3x + 9. Give the quotient and the remainder.

Step-by-step solution

1. 1

   Expect fractions in the quotient. The divisor leads with $3x$ while the dividend leads with $x^2$, so the first quotient term is $\dfrac{x^2}{3x}=\dfrac{x}{3}$. Do not factor the $3$ out of the divisor first — that would change what "remainder" means.

2. 2

   First round. Multiply back: $\dfrac{x}{3}(3x+9)=x^2+3x$. Subtract: $(x^2-12x+5)-(x^2+3x)=-15x+5.$

3. 3

   Second round. $\dfrac{-15x}{3x}=-5$. Multiply back: $-5(3x+9)=-15x-45$. Subtract: $(-15x+5)-(-15x-45)=50.$

4. 4

   Recognise the stopping point. The constant $50$ has lower degree than $3x+9$, so the quotient is $\dfrac{x}{3}-5$ and the remainder is $50$.

5. 5

   Check with the division identity. $\left(\frac{x}{3}-5\right)(3x+9)+50=x^2+3x-15x-45+50=x^2-12x+5.$