Factorize the displayed determinant.

01 / Standard#Your turn

D=\begin{vmatrix}a&a^2+1&bc\\b&b^2+1&ac\\c&c^2+1&ab\end{vmatrix}

Official paper · jm02-2021 · 5(a) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Subtract row 1 from rows 2 and 3.

Hint 2

Extract b−a, c−a and then c−b.

Worked solution

Row subtraction produces two polynomial factors.
$D=(b-a)(c-a)\begin{vmatrix}a&a^2+1&bc\\1&a+b&-c\\1&a+c&-b\end{vmatrix}$
Subtract the new row 2 from row 3, then extract c−b.
$D=(b-a)(c-a)(c-b)\begin{vmatrix}a&a^2+1&bc\\1&a+b&-c\\0&1&1\end{vmatrix}$
Subtract column 3 from column 2 and expand along the last row.
$D=(b-a)(c-a)(c-b)[a(a+b+c)-(a^2+1-bc)]$
Simplify the final factor and account for the two sign reversals.
$D=(a-b)(b-c)(c-a)(ab+bc+ca-1)$

D=(a−b)(b−c)(c−a)(ab+bc+ca−1).

Checks and common pitfalls: Factoring a polynomial identity does not require a,b,c to be distinct; equal-variable cases give determinant zero.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Row subtraction produces two polynomial factors.
$D=(b-a)(c-a)\begin{vmatrix}a&a^2+1&bc\\1&a+b&-c\\1&a+c&-b\end{vmatrix}$
Subtract the new row 2 from row 3, then extract c−b.
$D=(b-a)(c-a)(c-b)\begin{vmatrix}a&a^2+1&bc\\1&a+b&-c\\0&1&1\end{vmatrix}$
Subtract column 3 from column 2 and expand along the last row.
$D=(b-a)(c-a)(c-b)[a(a+b+c)-(a^2+1-bc)]$
Simplify the final factor and account for the two sign reversals.
$D=(a-b)(b-c)(c-a)(ab+bc+ca-1)$

Think first. Reveal a hint when the class is ready.

2021 JM02