Find the remainder after division by x+3.

01 / Standard#Your turn

P(x)=\sum_{j=0}^{1000}x^j

Official paper · jm01-2026 · I.5 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The remainder theorem substitutes the zero of the divisor.

Hint 2

The resulting sum is geometric with 1001 terms.

Worked solution

Substitute −3, not 3.
$R=P(-3)=\sum_{j=0}^{1000}(-3)^j$
Apply the finite geometric-sum formula.
$R=\frac{1-(-3)^{1001}}{1-(-3)}=\frac{1+3^{1001}}4$

D: (1+3¹⁰⁰¹)/4.

Checks and common pitfalls: Degree 1000 gives 1001 terms because the constant term is included.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Substitute −3, not 3.
$R=P(-3)=\sum_{j=0}^{1000}(-3)^j$
Apply the finite geometric-sum formula.
$R=\frac{1-(-3)^{1001}}{1-(-3)}=\frac{1+3^{1001}}4$

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

2026 JM01