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How do you split p(x)/[(x − a)(x − b)] (two different linear factors)?
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All Flashcards in Topic 1.11
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1.11.18 cards
How do you split p(x)/[(x − a)(x − b)] (two different linear factors)?
Write it as A/(x − a) + B/(x − b), then find A and B.
What is the cover-up method?
Clear the fractions, then substitute the x that makes one bracket zero — it deletes a term and leaves a single unknown.
To find A (the numerator over (x − a)), which x do you substitute?
x = a — the root of its own bracket — which zeroes the OTHER term and isolates A.
First step in any partial-fractions question?
Multiply both sides by the whole denominator to clear the fractions.
Split (5x − 1)/[(x + 1)(x − 2)].
2/(x + 1) + 3/(x − 2).
Split (x + 7)/[(x − 1)(x + 3)].
2/(x − 1) − 1/(x + 3).
When can you use the two-fraction split?
When the bottom is two DIFFERENT linear factors, e.g. (x − 1)(x + 3).
How do you check your A and B?
Add the two fractions back over a common denominator — you should recover the original fraction.
1.11.28 cards
What if the denominator is given as a quadratic, not two brackets?
Factorise it first into (x − a)(x − b), then split as usual.
When is a fraction 'top-heavy' (improper)?
When the top's highest power is as big as (or bigger than) the bottom's. You must divide first.
How do you handle a top-heavy fraction?
Divide to peel off a whole part, leaving a proper fraction; then split the proper part into partial fractions.
Split x²/[(x − 1)(x + 1)].
1 + (1/2)/(x − 1) − (1/2)/(x + 1) — divide first since x² = (x² − 1) + 1.
Factorise x² + x − 6.
(x + 3)(x − 2).
Factorise x² − 4 to split a fraction over it.
(x − 2)(x + 2) — difference of two squares.
Besides cover-up, how else can you find A and B?
Equate coefficients: expand the right side and match the x-terms and the constant terms, then solve.
Can you split (x² + 1)/(x² − 1) straight away?
No — same degree top and bottom (top-heavy). Divide first: it's 1 + 2/(x² − 1).
Topic 1.11 study notes
Full notes & explanations for Partial fractions (HL only)
Math AA exam skills
Paper structures, command terms & tips
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