Dividing by a quadratic

6 minA Level3.1.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can divide a quartic by a quadratic, write the answer as $\text{quotient} + \dfrac{\text{remainder}}{\text{divisor}}$ , and check it.

What changes when the divisor is quadratic

Two things change, and both follow from one rule: stop when the leftover degree is smaller than the divisor's degree.

The quotient is two degrees lower than the dividend. A quartic divided by a quadratic gives a quadratic quotient.
The remainder can now be linear, $px + q$ , because a leftover of degree 1 is already smaller than the divisor's degree 2.

In symbols, for a quartic divided by a quadratic:

$P(x) = (ax^2 + bx + c)Q(x) + (px + q)$

In words: the quotient is what divides exactly, and the linear piece is what cannot be divided any further.

Key to the symbols

Symbol	What it means here
$P(x)$	the dividend, a polynomial of degree up to 4
$ax^2 + bx + c$	the quadratic divisor
$Q(x)$	the quotient; two degrees lower than $P$
$px + q$	the remainder, of degree at most 1

The method

Set out the dividend in descending powers, with zeros for missing terms.
Divide the leading terms to get the first quotient term.
Multiply the whole quadratic by that term.
Subtract the product from what is inside.
Repeat until the leftover has degree 1 or 0 — that leftover is the remainder.
Check by expanding, or by substituting two convenient values of $x$ .

The forgotten step: subtracting a quadratic multiple each time. Multiplying only by the leading term of the divisor leaves the rest of the product unsubtracted, and every later line is then wrong.

Worked example: Example 1: a quartic by a quadratic

Divide $x^4 + 3x^2 - 2x + 1$ by $x^2 + 1$ .

Have a go first?

Quick check

CalculateQuestion 1 · Apply

Answer the 3 questions above to continue.