IGCSE Additional Mathematics: Factors of Polynomials Practice Questions
The remainder theorem states that dividing f(x) by (x minus a) leaves a remainder of f(a). The factor theorem is the special case where f(a) equals zero, meaning (x minus a) is a factor.
Topic 5 of Cambridge IGCSE Additional Mathematics 0606 is highly mechanical once the two theorems are learned. The step most often skipped is stating that the remainder is zero before concluding a factor. The questions below cover remainders, factors, unknown coefficients and full cubic solutions.
What you need to know for Factors of Polynomials
- Remainder theoremThe remainder when f(x) is divided by (x minus a) equals f(a). For a divisor (bx minus a), evaluate at x equal to a divided by b.
- Factor theoremIf f(a) = 0 then (x minus a) is a factor of f(x), and conversely.
- Finding an unknown coefficientSubstitute the value that makes the divisor zero, set the result equal to the given remainder, and solve.
- Factorising a cubicFind one root by trial from the factors of the constant term, divide out that factor, then factorise the remaining quadratic.
- Checking rootsTry the factors of the constant term first, both positive and negative. One of them is nearly always a root.
IGCSE Additional Mathematics Factors of Polynomials questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Find the remainder when f(x) = x3 minus 2x2 + 3x minus 5 is divided by (x minus 2).
Show the worked answer
- By the remainder theorem, the remainder equals f(2).
- f(2) = 23 minus 2 x 22 + 3 x 2 minus 5.
- = 8 minus 8 + 6 minus 5 = 1. The remainder is 1, so (x minus 2) is not a factor.
Show that (x minus 1) is a factor of f(x) = x3 + 2x2 minus 5x + 2, and factorise f(x) as far as possible.
Show the worked answer
- Evaluate f(1) = 1 + 2 minus 5 + 2 = 0.
- Since f(1) = 0, by the factor theorem (x minus 1) is a factor.
- Divide f(x) by (x minus 1), by long division or by comparing coefficients, giving x2 + 3x minus 2.
- So f(x) = (x minus 1)(x2 + 3x minus 2).
- The quadratic does not factorise over the integers, since its discriminant is 9 + 8 = 17, which is not a perfect square. This is the fully factorised form.
Given that (x + 2) is a factor of f(x) = x3 + ax2 + x + 6, find the value of a.
Show the worked answer
- If (x + 2) is a factor then f(minus 2) = 0.
- f(minus 2) = (minus 2)3 + a(minus 2)2 + (minus 2) + 6.
- = minus 8 + 4a minus 2 + 6 = 4a minus 4.
- Set equal to zero: 4a minus 4 = 0, so a = 1.
Solve the equation x3 minus 6x2 + 11x minus 6 = 0.
Show the worked answer
- Try factors of the constant term 6. f(1) = 1 minus 6 + 11 minus 6 = 0, so (x minus 1) is a factor.
- Divide to obtain the quadratic factor: x2 minus 5x + 6.
- So the equation becomes (x minus 1)(x2 minus 5x + 6) = 0.
- Factorise the quadratic: (x minus 2)(x minus 3).
- The full factorisation is (x minus 1)(x minus 2)(x minus 3) = 0, giving x = 1, 2 or 3.
Common mistakes in this topic
- Using the wrong sign when substituting for a factor of the form (x + a).
- Concluding a factor without stating that the remainder is zero.
- Forgetting brackets when substituting a negative value into an even power.
- Stopping after finding one root of a cubic.
- Testing values that are not factors of the constant term.
Exam tips
- Set the divisor equal to zero to find which value to substitute. It removes every sign error.
- State the conclusion explicitly: since f(a) = 0, (x minus a) is a factor. That sentence carries a mark.
- For a cubic, test the factors of the constant term first, positive and negative.
- After dividing out one factor, always try to factorise the remaining quadratic.
- Check your factorisation by expanding it back or by verifying another root.
Practise 20 more questions like this, free
vStudyWise marks every answer instantly, tracks the topics you keep dropping marks on and turns them into a weekly study plan.
Factors of Polynomials FAQs
What is the remainder theorem?
When a polynomial f(x) is divided by (x minus a), the remainder equals f(a). For a divisor of the form (bx minus a), substitute x equals a divided by b. Setting the divisor equal to zero always tells you which value to use.
What is the factor theorem?
If f(a) equals zero then (x minus a) is a factor of f(x), and if (x minus a) is a factor then f(a) equals zero. It is the special case of the remainder theorem where the remainder is zero, and it is the standard way to start factorising a cubic.
How do I factorise a cubic?
Test the factors of the constant term, both positive and negative, until one gives zero. That identifies a linear factor. Divide the cubic by it, using long division or comparison of coefficients, then factorise the resulting quadratic if possible.
How do I find an unknown coefficient?
Substitute the value that makes the divisor zero into the polynomial, set the result equal to the given remainder, or to zero if it is stated to be a factor, then solve the resulting equation for the unknown.
Continue through the IGCSE Additional Mathematics syllabus
Related IGCSE Additional Mathematics topics
See all 17 IGCSE Additional Mathematics practice topics ›
Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.