Papers 2 and 3. Divide a polynomial of degree up to 4 by a linear or quadratic divisor, find quotient and remainder, and use the factor and remainder theorems to factorise, solve equations and find unknown coefficients.
\(p(x) = (\text{divisor}) \times (\text{quotient}) + \text{remainder}\), where the remainder has lower degree than the divisor. The remainder when \(p(x)\) is divided by \((ax + b)\) is \(p\left(-\dfrac{b}{a}\right)\), and \((ax + b)\) is a factor exactly when \(p\left(-\dfrac{b}{a}\right) = 0\).
Fig. 1: p(x) = 2x³ + x² - 13x + 6 crosses the x-axis at -3, 0.5 and 2, so (x + 3), (2x - 1) and (x - 2) are its factors
Section 01
Algebraic long division.
Polynomial division works like numerical long division. Divide the leading term, multiply back, subtract, bring down, repeat. Stop when what is left has a lower degree than the divisor: that is the remainder.
The identity: \(p(x) \equiv d(x)\,q(x) + r(x)\), where \(d\) is the divisor, \(q\) the quotient, \(r\) the remainder.
Degrees: dividing a degree-\(n\) polynomial by a linear divisor gives a quotient of degree \(n - 1\) and a constant remainder; by a quadratic divisor, a quotient of degree \(n - 2\) and a remainder of the form \(ax + b\).
Missing powers: write them in with a zero coefficient (\(0x^2\)) so the columns line up.
Worked example Dividing by a linear divisor
Find the quotient and remainder when \(3x^3 - 4x^2 + 5x - 7\) is divided by \((x - 2)\).
Subtracting a negative term is where marks go: \((2x^2 + 5x) - (2x^2 - 4x) = 9x\), not \(x\). Put the line you subtract in brackets and change every sign.
Section 02
The remainder and factor theorems.
Put \(x = -\dfrac{b}{a}\) into \(p(x) \equiv (ax + b)\,q(x) + r\): the bracket becomes zero, leaving \(r\). So you can find a remainder without dividing, and a zero remainder means you have found a factor.
Divisor
Substitute
Remainder
Factor when
\(x - c\)
\(x = c\)
\(p(c)\)
\(p(c) = 0\)
\(x + c\)
\(x = -c\)
\(p(-c)\)
\(p(-c) = 0\)
\(ax + b\)
\(x = -\dfrac{b}{a}\)
\(p\left(-\dfrac{b}{a}\right)\)
\(p\left(-\dfrac{b}{a}\right) = 0\)
Worked example Remainder for a non-unit divisor
Find the remainder when \(4x^3 - 2x^2 + 3x + 1\) is divided by \((2x + 1)\).
Check the middle terms: \(2x \cdot x - 1 \cdot x^2 = x^2\) and \(2x(-6) - 1(x) = -13x\). Correct.
Factorise the quadratic: \(x^2 + x - 6 = (x + 3)(x - 2)\).
Answer: \(p(x) = (2x - 1)(x + 3)(x - 2)\), so \(x = \dfrac{1}{2},\ -3\) or \(2\) (Fig. 1)
Your turn
Find the remainder when \(x^3 + 4x^2 - 7\) is divided by \((x + 3)\).
\(p(-3) = -27 + 36 - 7 = 2\). Remainder \(2\).
Examiner trap
For the divisor \((x + 3)\) you substitute \(x = -3\), not \(x = 3\); for \((2x - 1)\) you substitute \(x = \dfrac{1}{2}\), not \(x = 1\) or \(x = 2\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
What is the remainder when \(x^3 - 2x + 5\) is divided by \((x - 1)\)?
Quick check+10 XP
To test whether \((3x - 2)\) is a factor of \(p(x)\), you evaluate
Quick check+10 XP
A polynomial of degree 4 is divided by a quadratic. What is the most the degree of the remainder can be?
Quick check+10 XP
\(p(x) = x^3 + kx - 6\) has \((x - 3)\) as a factor. What is \(k\)?
Section 03
Finding unknown coefficients.
Each piece of information (a factor, or a given remainder) gives one equation in the unknowns. Two unknowns need two facts: write both equations, then solve them simultaneously.
Worked example Two conditions, two unknowns
The polynomial \(p(x) = 2x^3 + ax^2 + bx - 6\) has \((x - 2)\) as a factor, and leaves remainder \(-12\) when divided by \((x + 1)\). Find \(a\) and \(b\).
Factor theorem: \(p(2) = 16 + 4a + 2b - 6 = 0\), so \(4a + 2b = -10\), i.e. \(2a + b = -5\).
Remainder theorem: \(p(-1) = -2 + a - b - 6 = -12\), so \(a - b = -4\).
Add the equations: \(3a = -9\), so \(a = -3\).
Then \(b = a + 4 = 1\).
Answer: \(a = -3,\ b = 1\)
Worked example Then factorise fully
With \(a = -3\) and \(b = 1\), show that \(p(x) = 0\) has only one real root.
\(p(x) = 2x^3 - 3x^2 + x - 6\). Since \((x - 2)\) is a factor, \(p(x) = (x - 2)(2x^2 + cx + 3)\).