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Polynomials, factor and remainder theorems

Revision notes · spec MM1.6

Section 1 Part 1: AS-level pure mathematics
✎ 17 practice questions⚡ Quick-fireFormula cards

The Factor Theorem and Remainder Theorem convert 'is \((x - a)\) a factor?' into 'is \(f(a) = 0\)?'. Combined with comparing coefficients this handles almost every TMUA polynomial question, including those with divisors of the form \(ax + b\) or a quadratic.

Contents: Factor and Remainder Theorems (MM1.6)Factorising and rootsMust-knowTrapsReal questions

Factor and Remainder Theorems (MM1.6)

  • Remainder Theorem: when \(f(x)\) is divided by \((x - a)\) the remainder is \(f(a)\); for \((ax - b)\) it is \(f\!\left(\frac ba\right)\).
  • Factor Theorem: \((x - a)\) is a factor of \(f(x)\) if and only if \(f(a) = 0\).
  • Dividing by a quadratic leaves a remainder of degree at most \(1\): write \(f(x) = (x^2 - 1)q(x) + Ax + B\) and substitute the roots \(x = \pm1\) to find \(A\) and \(B\).
  • To find a quotient quickly, compare coefficients: \(2x^3 - 3x^2 + 4x - 5 = (x - 1)(2x^2 + px + q) + r\).

Factorising and roots

  • A cubic \(x^3 + ax^2 + bx + c\) with roots \(p, q, r\) equals \((x - p)(x - q)(x - r)\), so \(p + q + r = -a\), \(pq + qr + rp = b\), \(pqr = -c\).
  • \(f(1)\) is the sum of the coefficients; \(f(-1)\) is the alternating sum.
  • Useful identities: \(x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2)\) (add and subtract \(4x^2\)); \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
  • A repeated factor \((x - a)^2\) means \(f(a) = 0\) and \(f'(a) = 0\) (the graph touches the axis).

Must-know (no formula booklet)

  • remainder on dividing by \((ax - b)\) is \(f(b/a)\)
  • \((x - a)\) factor \(\Leftrightarrow f(a) = 0\)
  • degree of remainder \(<\) degree of divisor
  • \(f(1) =\) sum of coefficients

Classic traps

  • For divisor \((2x - 1)\) substitute \(x = \frac12\), not \(x = 1\) or \(x = 2\).
  • Not every cubic factorises over the integers - test \(\pm\) factors of the constant term systematically.
  • Comparing coefficients needs the same number of equations as unknowns; a leftover equation is a check, not a contradiction.

Worked example

Q. \(x^3 + px + q\) is divisible by \((x - 2)^2\). Find \(q\).

A. \(x^3 + px + q = (x - 2)^2(x - r)\); the \(x^2\) coefficient gives \(-4 - r = 0\), so \(r = -4\) and \(q = (4)(4) = 16\).

How it has been examined

2023 · Paper 1Question 7polynomials P(x) and Q(x)question papersit online
2023 · Paper 2Question 19roots of f(x) and x f(x)question papersit online
2020 · Paper 1Question 2factors of 2x^3 + px^2 + qquestion papersit online
2020 · Paper 1Question 20(x - a)(x^2 - x + a) = 0: exactly two rootsquestion papersit online
2020 · Paper 2Question 18f - g > 0 for cubicsquestion papersit online
2019 · Paper 2Question 2factors of 2x^3 + px^2 + qquestion papersit online
October 2018 · Paper 1Question 5cubic with coefficients 1, 2, 3 in some orderquestion papersit online
November 2017 · Paper 1Question 4remainder theorem with (px - 1)question papersit online
Practice papers · Paper 1Question 2factorise a cubic with factor (x + 2)question papersit online
Specimen papers · Paper 1Question 6(x - 2) is a factor: sum of possible valuesquestion papersit online
Specimen papers · Paper 1Question 11sum of roots of a factorised quarticquestion papersit online
Specimen papers · Paper 2Question 12deductions from p(1) = 2question papersit online