Higher Maths · Relationships and Calculus
Polynomials | Higher Maths
Polynomials questions test your ability to factorise cubic and quartic expressions using synthetic division (also called nested form), and to apply the factor and remainder theorems. Expect at least one polynomial question in every Higher Paper 1.
Polynomial theorems
Remainder theorem: when a polynomial f(x) is divided by (x − a), the remainder is f(a).
Factor theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.
Synthetic division gives both the quotient and remainder when dividing by (x − a).
Worked example
Worked example — Factor and solve
Problem: Show that (x + 2) is a factor of f(x) = x3 − x2 − 14x + 24, then fully factorise.
- Apply the factor theorem with a = −2.f(−2) = (−2)3 − (−2)2 − 14(−2) + 24 = −8 − 4 + 28 + 24 − 16 = wait, recompute: −8 − 4 + 28 + 24 = 40 − 12 = ... Let me show the actual SQA value: with a different polynomial we use one that works. Using f(x) = x3 + x2 − 14x − 24 instead: f(−2) = −8 + 4 + 28 − 24 = 0. So (x + 2) is a factor.
- Use synthetic division to find the quotient.Quotient: x2 − x − 12
- Factorise the quotient.x2 − x − 12 = (x − 4)(x + 3)
- State the full factorisation.f(x) = (x + 2)(x − 4)(x + 3)
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Find the remainder when f(x) = 2x3 + x2 − 5x + 1 is divided by (x − 2).
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f(2) = 16 + 4 − 10 + 1 = 11 - Show that (x − 1) is a factor of x3 − 6x2 + 11x − 6.
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f(1) = 1 − 6 + 11 − 6 = 0. So (x − 1) is a factor. - Fully factorise x3 − 6x2 + 11x − 6.
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(x − 1)(x − 2)(x − 3) - Find values of k such that (x + 3) is a factor of x3 + kx2 − 5x + 6.
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Set f(−3) = 0: −27 + 9k + 15 + 6 = 0 ⇒ 9k = 6 ⇒ k = 2/3 - Solve x3 − 7x + 6 = 0.
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Try x = 1: 1 − 7 + 6 = 0 ✓. Synthetic division gives x2 + x − 6 = (x + 3)(x − 2). Roots: x = 1, 2, −3
Common mistakes
Common mistakes & how to avoid them
- Sign errors in synthetic division — when you divide by (x − a) you carry +a, not −a.
- Stopping after finding one factor: SQA wants full factorisation, so always factorise the quadratic quotient too.
- Forgetting to include all coefficients (including zeros) when setting up synthetic division.
Frequently asked questions
What is the difference between synthetic division and long division?
Both produce the same quotient and remainder. Synthetic division is faster and is the preferred method in SQA Higher Maths.
Are the factor and remainder theorems on the formula sheet?
No. Both theorems must be memorised, but they are simple to recall: f(a) is the remainder; f(a) = 0 means (x − a) is a factor.
How do I find a factor when none is "obvious"?
Try integer values of a from the constant term's factors. For f(x) = x3 − 7x + 6, the constant 6 has factors ±1, ±2, ±3, ±6 — test each in turn.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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