Factor & remainder theorems
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All 8 Flashcards — Factor & remainder theorems
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Question
State the remainder theorem.
Answer
The remainder when P(x) is divided by (x − a) is P(a).
Question
State the factor theorem.
Answer
(x − a) is a factor of P(x) if and only if P(a) = 0.
Question
How do you find the remainder on dividing by (x − a)?
Answer
Substitute: the remainder is P(a) — no long division needed.
Question
How does a given factor or remainder help find unknowns?
Answer
It gives an equation (P(value) = 0 for a factor, or = remainder); solve the equations together.
Question
What does (x − a)² being a factor require?
Answer
Both P(a) = 0 and P′(a) = 0 (a is a repeated root).
Question
Remainder when x³ − 2x² + 5x − 1 is divided by (x − 2)?
Answer
P(2) = 8 − 8 + 10 − 1 = 9.
Question
Is (x − 1) a factor of x³ − 6x² + 11x − 6?
Answer
P(1) = 1 − 6 + 11 − 6 = 0, so yes.
Question
Divide by (x + 2): which value do you substitute?
Answer
x = −2 (the root of x + 2).
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Full study notes for Factor & remainder theorems
Topic 2.12 hub
Factor & remainder (HL only)
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