Algebra · Junior Cert

Multiplying Out

Junior Cert Higher · Single & double brackets, the grid · Tap NEXT to begin

Section 1 of 4

Simple Multiplying

To multiply, multiply the numbers and collect the letters. A number outside a bracket multiplies every term inside.

Worked examples

$2(5) = 10$

$3(2x) = 6x$

$2a(5b) = 10ab$

$2x(3x) = 6x^2$

Worked examples — one bracket

$3(x+2) = 3x+6$

$5(2x+3y) = 10x+15y$

$3(5p+2q) = 15p+6q$

Worked examples — a letter outside

$x(x+3) = x^2+3x$

$x(2x+5) = 2x^2+5x$

The grid (area) model shows why $x(x+3)=x^2+3x$:

You try

Multiply out $3(2p+q)$.

Pen and paper out — try it before you reveal.

Multiply each term by $3$.

$6p+3q$

You try

Multiply out $2x(3x+5)$.

Pen and paper out — try it before you reveal.

$2x\cdot 3x = 6x^2$, $2x\cdot 5 = 10x$

$6x^2+10x$

Section 2 of 4

Building Two Brackets

A product like $(x+5)(x+3)$ is done by splitting: multiply the whole second bracket by the first term, then by the second term.

Worked example — the split method

Simplify $x(x+5)+3(x+5)$.

$x^2+5x+3x+15$

$x^2+8x+15$

Watch the signs: like signs give $+$, unlike signs give $-$.

Worked example

Simplify $x(x-5)-3(x-5)$.

$x^2-5x-3x+15$

$x^2-8x+15$

Worked example

Simplify $x(x-7)-5(x-7)$.

$x^2-7x-5x+35$

$x^2-12x+35$

Worked example

Simplify $x(x+7)+3(x+7)$.

$x^2+7x+3x+21$

$x^2+10x+21$

Worked example

Simplify $x(2x+1)+5(2x+1)$.

$2x^2+x+10x+5$

$2x^2+11x+5$

Worked example

Simplify $5x(3x-1)+4(3x-1)$.

$15x^2-5x+12x-4$

$15x^2+7x-4$

You try

Simplify $3x(2x-5)-4(2x-5)$.

Pen and paper out — try it before you reveal.

$6x^2-15x-8x+20$

$6x^2-23x+20$

Section 3 of 4

Full Brackets & the Grid

Now with both brackets written out. The area grid gives the same four pieces.

Worked example — find the area

Find the area $(x+2)(x+3)$.

$x^2+2x+3x+6$

$x^2+5x+6$

Worked example

Simplify $(x+5)(x+4)$.

$x(x+4)+5(x+4)$

$x^2+4x+5x+20$

$x^2+9x+20$

Worked example

Simplify $(x+2)(x+9)$.

$x(x+9)+2(x+9)$

$x^2+9x+2x+18$

$x^2+11x+18$

Worked example

Simplify $(2x+3)(5x+6)$.

$2x(5x+6)+3(5x+6)$

$10x^2+12x+15x+18$

$10x^2+27x+18$

Worked example

Simplify $(3x+1)(5x+7)$.

$3x(5x+7)+1(5x+7)$

$15x^2+21x+5x+7$

$15x^2+26x+7$

You try

Simplify $(5x+3)(6x+7)$.

Pen and paper out — try it before you reveal.

$5x(6x+7)+3(6x+7)$

$30x^2+35x+18x+21$

$30x^2+53x+21$

You try

Simplify $(5x+3)(2x+7)$.

Pen and paper out — try it before you reveal.

$5x(2x+7)+3(2x+7)$

$10x^2+35x+6x+21$

$10x^2+41x+21$

You try

Simplify $(7x+3)(6x+5)$.

Pen and paper out — try it before you reveal.

$7x(6x+5)+3(6x+5)$

$42x^2+35x+18x+15$

$42x^2+53x+15$

You try

Simplify $(5x+2)(6x+8)$.

Pen and paper out — try it before you reveal.

$5x(6x+8)+2(6x+8)$

$30x^2+40x+12x+16$

$30x^2+52x+16$

Section 4 of 4

Brackets With Minus Signs

The mode is the same — multiply every term — but track the signs. A plus means ‘I have’, a minus means ‘I owe’.

Worked example

Simplify $(3x-2)(5x+4)$.

$3x(5x+4)-2(5x+4)$

$15x^2+12x-10x-8$

$15x^2+2x-8$

Worked example

Simplify $(3x-7)(2x-5)$.

$3x(2x-5)-7(2x-5)$

$6x^2-15x-14x+35$

$6x^2-29x+35$

You try

Simplify $(5x-3)(7x-2)$.

Pen and paper out — try it before you reveal.

$5x(7x-2)-3(7x-2)$

$35x^2-10x-21x+6$

$35x^2-31x+6$

You try

Simplify $(5x-3)(4x-7)$.

Pen and paper out — try it before you reveal.

$5x(4x-7)-3(4x-7)$

$20x^2-35x-12x+21$

$20x^2-47x+21$

You try

Simplify $(2x+3)(5x+2)$.

Pen and paper out — try it before you reveal.

$2x(5x+2)+3(5x+2)$

$10x^2+4x+15x+6$

$10x^2+19x+6$

You try

Simplify $(3x-2)(6x-5)$.

Pen and paper out — try it before you reveal.

$3x(6x-5)-2(6x-5)$

$18x^2-15x-12x+10$

$18x^2-27x+10$

That’s expanding.

Every term times every term, mind the signs, and collect the like terms.