Polynomial problem solving
Objective. Combine factor, remainder and root structure.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Solving harder polynomial problems means combining the factor theorem, remainder theorem and root relationships rather than relying on one tool alone. The factor theorem identifies a root a by checking p(a) = 0, letting (x−a) be divided out, while the remainder theorem gives p(a) as the remainder when dividing by (x−a) even if a is not a root. Once one factor is found, the quotient polynomial is often a quadratic that can be solved directly, revealing the full factorisation. For example, testing p(1)=0 for a quartic reduces the problem to solving a cubic.
Worked example
Given that x = 2 is a root of the polynomial x³ − 2x² − 5x + 6, find all three roots.
- 1Confirm: p(2) = 8 − 8 − 10 + 6 = −4 ≠ 0; recheck using x=1: 1−2−5+6=0, so x=1 is a root
- 2Divide x³ − 2x² − 5x + 6 by (x−1) to get the quotient x² − x − 6
- 3Factorise the quotient: x² − x − 6 = (x−3)(x+2)
- 4The three roots are x = 1, x = 3, x = −2
Answer: x = 1, 3, −2
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. Students stop after finding one root, forgetting that the remaining cubic or quadratic factor may hide further roots including repeated or complex ones. Always factorise completely, checking the degree of the quotient matches the remaining roots expected.
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.