Dividing polynomials

Warm-up

Three quick questions from earlier lessons.

Discover

$x^2 + 5x + 6 = (x + 2)(\ ?\ )$ . What goes in the gap?

$x + 3$
$x + 2$

Key idea

Long division works for polynomials too.

Divide the leading term, multiply back, subtract, bring down, repeat. Write missing powers with coefficient 0. Result:

p(x) = d(x)\,q(x) + r(x)

, with the remainder

r

of lower degree than

d

. Dividing by

x - a

, the remainder is

p(a)

Worked example

$(x^3 - 8) \div (x - 2)$ .

2x^2 + 0x - 8

x³ ÷ x = x²; subtract x²(x − 2) = x³ − 2x².

4x - 8

2x² ÷ x = 2x; subtract 2x(x − 2).

x^3 - 8 = (x - 2)(x^2 + 2x + 4)

4x ÷ x = 4; remainder 0.

Your turn, guided

Your turn: the remainder of $(x^2 + x + 1) \div (x - 1)$ .

3

Evaluate at 1.

Practice ★

Divide: $\dfrac{x^2 + 13x + 42}{x + 6}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 12x + 35}{x + 7}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 10x + 16}{x + 2}$ . Enter the quotient.

Watch out!

Practice ★★

Divide $x^3 - 1$ by $x - 1$ . Enter the quotient.
Divide $x^3 - 27$ by $x - 3$ . Enter the quotient.

Find the remainder when $x^2 + 4x + 1$ is divided by $x - 4$ .

Boss problem. $p(x) = x^3 - 8x^2 + 17x - 10$ has the factor $x - 1$ .

Exit ticket

Three questions to check you have got it.

Divide: $\dfrac{x^2 + 13x + 42}{x + 6}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 12x + 35}{x + 7}$ . Enter the quotient.
Divide: $\dfrac{x^2 + 10x + 16}{x + 2}$ . Enter the quotient.

Summary card

Polynomial division

$p = d \cdot q + r$ ; keep 0-coefficients for missing powers.

Watch out: check by multiplying back.

End of the lesson

Next lesson: The remainder theorem and the factor theorem Back to the chapter

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