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Polynomial division

Polynomial division is the same idea as the division algorithm for numbers: divide the leading term, multiply back by the divisor, subtract, and repeat until the degree is lower than that of the divisor. The remainder theorem tells you when the division is exact. A cubic with one known zero then factorises into a linear factor times a quadratic, which you factorise further.

P(x)=(x−a)Q(x)P(x) = (x - a)Q(x)when P(a)=0P(a) = 0
x3−2x2−x+2=(x−1)(x−2)(x+1)x^3 - 2x^2 - x + 2 = (x - 1)(x - 2)(x + 1)complete factorisation

Symbols

Q(x)Q(x)the quotient
aaa zero

Example

P(x)=x3−2x2−x+2P(x) = x^3 - 2x^2 - x + 2. P(1)=1−2−1+2=0P(1) = 1 - 2 - 1 + 2 = 0, so P(x)=(x−1)(x2−x−2)=(x−1)(x−2)(x+1)P(x) = (x - 1)(x^2 - x - 2) = (x - 1)(x - 2)(x + 1). The zeros are 11, 22 and −1-1.

Check the other zeros by substituting them into P(x)P(x).
Practise factorisation and polynomial division for free →

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Part of Mathematics 1T: Factorisation and polynomial division.