Factor the determinant.

01 / Standard#Your turn

D=\begin{vmatrix}a&b+c&b^2+c^2\\b&a+c&a^2+c^2\\c&a+b&a^2+b^2\end{vmatrix}

Official paper · jm02-2022 · 5(a) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Let s=a+b+c and t=a²+b²+c².

Hint 2

Use column operations to reduce to a Vandermonde determinant.

Worked solution

Let v=(a,b,c)ᵀ and v^[2]=(a²,b²,c²)ᵀ. Add column 1 to column 2 and use multilinearity to remove the repeated constant column, without division.
$D=\det[\,v,\ s\mathbf1,\ t\mathbf1-v^{[2]}\,]=-s\det[\,v,\mathbf1,v^{[2]}\,]=s\det[\,\mathbf1,v,v^{[2]}\,]$
Evaluate the Vandermonde determinant and reorder factors.
$D=(a+b+c)(b-a)(c-a)(c-b)=(a-b)(b-c)(c-a)(a+b+c)$

(a−b)(b−c)(c−a)(a+b+c).

Checks and common pitfalls: The derivation does not divide by a+b+c, so it also covers a+b+c=0.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Let v=(a,b,c)ᵀ and v^[2]=(a²,b²,c²)ᵀ. Add column 1 to column 2 and use multilinearity to remove the repeated constant column, without division.
$D=\det[\,v,\ s\mathbf1,\ t\mathbf1-v^{[2]}\,]=-s\det[\,v,\mathbf1,v^{[2]}\,]=s\det[\,\mathbf1,v,v^{[2]}\,]$
Evaluate the Vandermonde determinant and reorder factors.
$D=(a+b+c)(b-a)(c-a)(c-b)=(a-b)(b-c)(c-a)(a+b+c)$

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

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