Algebra · Junior Cert

Long Division

Junior Cert Higher · Dividing quadratics & cubics · Tap NEXT to begin

Section 1 of 3

Dividing a Quadratic

Polynomial long division works just like number division: divide, multiply, subtract, bring down — and repeat.

Worked example

Divide $x^2+8x+12$ by $x+2$.

$x^2 \div x = x$. Then $x(x+2) = x^2+2x$.

Subtract: $8x-2x = 6x$, bring down $+12 \Rightarrow 6x+12$.

$6x \div x = 6$. Then $6(x+2) = 6x+12$. Subtract: remainder $0$.

$x+6$

Worked example

Divide $x^2+7x+12$ by $x+3$.

$x(x+3) = x^2+3x \Rightarrow$ leaves $4x+12$.

$4(x+3) = 4x+12 \Rightarrow$ remainder $0$.

$x+4$

Worked example

Divide $x^2+9x+20$ by $x+4$.

$x(x+4)=x^2+4x \Rightarrow 5x+20$; $5(x+4)=5x+20$.

$x+5$

Worked example — with a minus

Divide $x^2-11x+30$ by $x-6$.

$x(x-6)=x^2-6x \Rightarrow -5x+30$; $-5(x-6)=-5x+30$.

$x-5$

You try

Divide $x^2+13x+42$ by $x+6$.

Pen and paper out — try it before you reveal.

$x(x+6)=x^2+6x \Rightarrow 7x+42$

$7(x+6)=7x+42 \Rightarrow$ remainder $0$

$x+7$

Worked example

Divide $x^2-15x+50$ by $x-5$.

$x(x-5)=x^2-5x \Rightarrow -10x+50$; $-10(x-5)=-10x+50$.

$x-10$

Section 2 of 3

When the Front Isn’t 1

Same method — just divide by the leading term each time.

Worked example

Divide $6x^2+11x-35$ by $2x+7$.

$6x^2 \div 2x = 3x$. $3x(2x+7)=6x^2+21x \Rightarrow -10x-35$.

$-10x \div 2x = -5$. $-5(2x+7)=-10x-35 \Rightarrow$ remainder $0$.

$3x-5$

Section 3 of 3

Dividing a Cubic

With an $x^3$ term the quotient starts with $x^2$. Keep going until nothing is left.

Worked example

Divide $x^3+3x^2+5x+3$ by $x+1$.

$x^2(x+1)=x^3+x^2 \Rightarrow 2x^2+5x$

$2x(x+1)=2x^2+2x \Rightarrow 3x+3$

$3(x+1)=3x+3 \Rightarrow$ remainder $0$

$x^2+2x+3$

Worked example

Divide $x^3+5x^2+11x+10$ by $x+2$.

$x^2(x+2)=x^3+2x^2 \Rightarrow 3x^2+11x$

$3x(x+2)=3x^2+6x \Rightarrow 5x+10$; $5(x+2)=5x+10$.

$x^2+3x+5$

Worked example

Divide $x^3+3x^2-10x-24$ by $x-3$.

$x^2(x-3)=x^3-3x^2 \Rightarrow 6x^2-10x$

$6x(x-3)=6x^2-18x \Rightarrow 8x-24$; $8(x-3)=8x-24$.

$x^2+6x+8$

You try

Divide $x^3+9x^2+23x+15$ by $x+3$.

Pen and paper out — try it before you reveal.

$x^2(x+3)=x^3+3x^2 \Rightarrow 6x^2+23x$

$6x(x+3)=6x^2+18x \Rightarrow 5x+15$

$5(x+3)=5x+15$

$x^2+6x+5$

Worked example

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

$x^2(x-1)=x^3-x^2 \Rightarrow 3x^2-x$

$3x(x-1)=3x^2-3x \Rightarrow 2x-2$; $2(x-1)=2x-2$.

$x^2+3x+2$

Worked example — a bigger leading term

Divide $2x^3-15x^2+34x-24$ by $2x-3$.

$2x^3 \div 2x = x^2$. $x^2(2x-3)=2x^3-3x^2 \Rightarrow -12x^2+34x$

$-6x(2x-3)=-12x^2+18x \Rightarrow 16x-24$; $8(2x-3)=16x-24$.

$x^2-6x+8$

Worked example — a missing term

Divide $x^3-2x-4$ by $x-2$. Write in the missing $x^2$ as $0x^2$: $x^3+0x^2-2x-4$.

$x^2(x-2)=x^3-2x^2 \Rightarrow 2x^2-2x$

$2x(x-2)=2x^2-4x \Rightarrow 2x-4$; $2(x-2)=2x-4$.

$x^2+2x+2$

That’s long division.

Divide, multiply, subtract, bring down — and mind any missing term with a $0$.