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Factorising a quadratic by inspection is realistic. Factorising by inspection is not — there are too many possibilities. The factor theorem gives you a way in: test values until one makes the polynomial zero, and you have found a factor with certainty rather than by search.
The big picture
This is the first genuinely result in the algebra chapter. Everything before it relied on recognising a pattern; this tells you what to do when you recognise nothing. The logic — a root at means a factor , and vice versa — connects three separate ideas that look different on the page: the roots of an equation, the factors of an expression, and the -intercepts of a curve. Once you see they are the same fact stated three ways, cubic sketching, polynomial equations and later work on partial fractions all become the same problem.
What you'll be able to do
If is a factor of a polynomial , then for some polynomial . Substituting makes the first bracket zero, so .
The converse also holds: if then must be a factor. Together they give a two-way test — and the useful direction is the one that lets you factors by hunting for roots.
Mind the sign. The factor corresponds to testing , and to testing . Substitute the value that makes the bracket vanish, not the number you can see written in it.
You are not testing at random. Any whole-number root of a polynomial with integer coefficients must divide the constant term exactly, so the candidate list is short and worth writing out before you start.
For , the constant is , so the candidates are . Small values first — is usually quickest to evaluate and surprisingly often works.
Tip — Write the conclusion in words — "since , is a factor". The final mark on "show that" questions is routinely for stating the reasoning, not for the arithmetic.
One root gives one linear factor. Dividing it out leaves a quadratic, which you already know how to handle — so a cubic never needs more than one application of the theorem.
You can divide formally, or compare coefficients: write and match terms. The leading coefficient and the constant come free, leaving one comparison to pin down the middle.
The word "hence" is an instruction. It means use the factorisation you have just produced — a solution that starts again from scratch may lose marks even if the roots are right.
The factor theorem is the special case of a more general result. Dividing by leaves a constant remainder, and that remainder is exactly .
So substituting tells you the remainder without doing any division at all. When the remainder is zero the division is exact — which is the factor theorem again, viewed from a different angle.
Tip — A question that asks for a remainder is telling you not to do long division. One substitution is the whole method.
Think like an examiner
Common misconceptions
Factor and remainder theorems
Stretch yourself
The polynomial has factor and leaves remainder when divided by . Find and .
Hint — Each condition gives one equation. The factor gives ; the remainder gives .
Questions students ask
Key takeaways
How this fits the course
Test yourself
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