Factor the determinant completely.

01 / Standard#Your turn

D=\begin{vmatrix}1&a&a^4\\1&b&b^4\\1&c&c^4\end{vmatrix}

Official paper · jm02-2026 · 5(a) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Subtract the first row from the other two.

Hint 2

Factor fourth-power differences and then the remaining difference in b and c.

Worked solution

After row subtraction, factor b−a and c−a from the last two rows.
$D=(b-a)(c-a)\big[(c^3+ac^2+a^2c+a^3)-(b^3+ab^2+a^2b+a^3)\big]$
Factor the bracket by c−b. The polynomial identity also holds when two parameters coincide.
$D=(b-a)(c-a)(c-b)(a^2+b^2+c^2+ab+ac+bc)$

(b−a)(c−a)(c−b)(a²+b²+c²+ab+ac+bc).

Checks and common pitfalls: Reversing one difference changes the sign; keep the row order consistent.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
After row subtraction, factor b−a and c−a from the last two rows.
$D=(b-a)(c-a)\big[(c^3+ac^2+a^2c+a^3)-(b^3+ab^2+a^2b+a^3)\big]$
Factor the bracket by c−b. The polynomial identity also holds when two parameters coincide.
$D=(b-a)(c-a)(c-b)(a^2+b^2+c^2+ab+ac+bc)$

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

2026 JM02