Finding all the roots & sketching
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Flip to reveal answersHow do you factorise a cubic fully?
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All 8 Flashcards — Finding all the roots & sketching
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Question
How do you factorise a cubic fully?
Answer
Find one root with the factor theorem, divide it out, then factorise/solve the resulting quadratic.
Question
Which trial values do you try for a root?
Answer
Small integers — factors of the constant term (±1, ±2, …).
Question
A real-coefficient polynomial has root a + bi. What else is a root?
Answer
The conjugate a − bi.
Question
How do you find the last real root once you have a complex pair?
Answer
Use the sum of roots (−b/a): subtract the known roots from it.
Question
Roots of x³ − 2x² − 5x + 6?
Answer
x = 1, 3, −2 (factorises as (x − 1)(x − 3)(x + 2)).
Question
Given 1 + i is a root of x³ − 4x² + 6x − 4, find the others.
Answer
1 − i (conjugate) and 2 (from sum of roots = 4).
Question
How does the leading term affect a polynomial sketch?
Answer
It sets the end behaviour: +xⁿ rises to the right; the parity of n sets the left end.
Question
How many real roots can a cubic have?
Answer
1 or 3 (complex roots come in pairs, and degree 3 is odd).
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Full study notes for Finding all the roots & sketching
Topic 2.12 hub
Factor & remainder (HL only)
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