Determine the three parameters in the factorisation using the leading coefficient, constant term and remainder.

01 / Standard#Your turn

f(x)=8x^4+ax^3+bx^2+cx+9=(px^2-3x+3)(2x^2+qx+r),\quad f(-1)=-10

Official paper · jm01-2025 · II.2(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Compare the x⁴ coefficient and the constant term.

Hint 2

Then substitute x=−1.

Worked solution

The extreme coefficients determine p and r.
$2p=8,\quad3r=9\implies p=4,\ r=3$
The remainder theorem determines q.
$f(-1)=(4+3+3)(2-q+3)=10(5-q)=-10\implies q=6$

p=4, q=6, r=3.

Checks and common pitfalls: Dividing by x+1 means substituting −1.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The extreme coefficients determine p and r.
$2p=8,\quad3r=9\implies p=4,\ r=3$
The remainder theorem determines q.
$f(-1)=(4+3+3)(2-q+3)=10(5-q)=-10\implies q=6$

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

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