Question

anja divides $(8x^3 - 36x^2 + 54x - 27)$ by $(2x - 3)$ as shown below. what error does anja make?
$\

$$\begin{array}{r}4x^2 - 12x + 8 \\\\ 2x - 3 \\enclose{longdiv}{8x^3 - 36x^2 + 54x - 27} \\\\ \\underline{8x^3 - 12x^2} \\\\ -24x^2 + 54x \\\\ \\underline{-24x^2 + 36x} \\\\ 18x - 27 \\\\ \\underline{18x + 27} \\\\ -54\\end{array}$$

$
she makes a subtraction error.
she makes an error choosing the $x$-term in the quotient.
she makes a multiplication error.
she makes an error rewriting the problem in long division.

Explanation:

Step1: Check final subtraction step

Anja subtracts $18x+27$ from $18x-27$, but the correct term to subtract is $(2x-3)\times9 = 18x-27$. The operation should be $(18x-27)-(18x-27)$, not $(18x-27)-(18x+27)$. This is a sign error in subtraction.

Step2: Verify other steps

• First quotient term: $\frac{8x^3}{2x}=4x^2$, $(2x-3)\times4x^2=8x^3-12x^2$, subtraction gives $-24x^2+54x$ (correct).
• Second quotient term: $\frac{-24x^2}{2x}=-12x$, $(2x-3)\times(-12x)=-24x^2+36x$, subtraction gives $18x-27$ (correct).
• Third quotient term: $\frac{18x}{2x}=9$, but Anja uses $18x+27$ instead of $18x-27$ for the subtraction step, which is a subtraction sign error.

Answer:

She makes a subtraction error.