Polynomial structure
Objective. Use factors, roots and identities to reason about polynomials.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Polynomial structure uses the relationships between a polynomial's factors, roots and coefficients to reason about its behaviour, for instance showing that if (x−a) is a factor then substituting x=a into the polynomial gives zero, known as the factor theorem. Identities such as the difference of two squares or sum/difference of cubes let polynomials be factorised or expanded without full long multiplication. The degree of a polynomial tells you the maximum number of real roots it can have, and the leading coefficient affects end behaviour. For example, since p(2)=0 for p(x)=x³−3x²+2, (x−2) must be a factor of p(x).
Worked example
Show that (x−1) is a factor of the polynomial p(x) = x³−2x²−x+2, then find the other factors.
- 1Substitute x=1 into p(x): 1−2−1+2 = 0, confirming (x−1) is a factor
- 2Divide p(x) by (x−1) to get the quotient x²−x−2
- 3Factorise the quotient: x²−x−2 = (x−2)(x+1)
- 4Write p(x) as the product of all three factors
Answer: p(x) = (x−1)(x−2)(x+1)
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Pupils sometimes assume a polynomial has as many roots as its degree even when some roots are not real, claiming a cubic must always cross the x-axis three times; a cubic always has at least one real root but may have only one if the others are complex.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.