Practice Flashcards
State the remainder theorem.
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All Flashcards in Topic 2.12
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2.12.18 cards
State the remainder theorem.
The remainder when P(x) is divided by (x − a) is P(a).
State the factor theorem.
(x − a) is a factor of P(x) if and only if P(a) = 0.
How do you find the remainder on dividing by (x − a)?
Substitute: the remainder is P(a) — no long division needed.
How does a given factor or remainder help find unknowns?
It gives an equation (P(value) = 0 for a factor, or = remainder); solve the equations together.
What does (x − a)² being a factor require?
Both P(a) = 0 and P′(a) = 0 (a is a repeated root).
Remainder when x³ − 2x² + 5x − 1 is divided by (x − 2)?
P(2) = 8 − 8 + 10 − 1 = 9.
Is (x − 1) a factor of x³ − 6x² + 11x − 6?
P(1) = 1 − 6 + 11 − 6 = 0, so yes.
Divide by (x + 2): which value do you substitute?
x = −2 (the root of x + 2).
2.12.28 cards
Sum and product of roots of ax² + bx + c = 0?
Sum = −b/a, product = c/a.
General sum and product of roots of a degree-n polynomial?
Sum = −aₙ₋₁/aₙ; product = (−1)ⁿ a₀/aₙ.
Cubic ax³ + bx² + cx + d: sum and product of roots?
Sum = −b/a; product = −d/a (the (−1)³ makes it negative).
Why use sum/product instead of solving?
It reads the symmetric functions of the roots straight off the coefficients — no need to find the roots.
Roots of 2x² − 6x + 1 = 0: sum and product?
Sum = 6/2 = 3, product = 1/2.
Roots of x³ − 4x² + x + 6 = 0: sum and product?
Sum = 4, product = −6.
Does the product of roots change sign with degree?
Yes — it's (−1)ⁿ a₀/aₙ, so + for even degree, − for odd.
Roots of x² − kx + (k+3) = 0 sum to 5 — find k.
Sum = k = 5.
2.12.38 cards
How do you factorise a cubic fully?
Find one root with the factor theorem, divide it out, then factorise/solve the resulting quadratic.
Which trial values do you try for a root?
Small integers — factors of the constant term (±1, ±2, …).
A real-coefficient polynomial has root a + bi. What else is a root?
The conjugate a − bi.
How do you find the last real root once you have a complex pair?
Use the sum of roots (−b/a): subtract the known roots from it.
Roots of x³ − 2x² − 5x + 6?
x = 1, 3, −2 (factorises as (x − 1)(x − 3)(x + 2)).
Given 1 + i is a root of x³ − 4x² + 6x − 4, find the others.
1 − i (conjugate) and 2 (from sum of roots = 4).
How does the leading term affect a polynomial sketch?
It sets the end behaviour: +xⁿ rises to the right; the parity of n sets the left end.
How many real roots can a cubic have?
1 or 3 (complex roots come in pairs, and degree 3 is odd).
Topic 2.12 study notes
Full notes & explanations for Factor & remainder (HL only)
Math AA exam skills
Paper structures, command terms & tips
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