Algebra · Junior Cert
Squares & Arrays
Junior Cert Higher · Perfect squares, trinomials & the array · Tap NEXT to begin
Section 1 of 5
Mixed Brackets
A quick warm-up: multiply every term, then collect like terms.
$2(3x) = 6x$
$5x(4x) = 20x^2$
$3(p+5) = 3p+15$
$q(2q+7) = 2q^2+7q$
Worked example — collect after expanding
Simplify $y(2y+3)+4(2y+3)$.
$2y^2+3y+8y+12$
$2y^2+11y+12$
Worked example
Simplify $2x(3x-5)-4(3x-5)$.
$6x^2-10x-12x+20$
$6x^2-22x+20$
Worked example — two separate brackets
Simplify $3(x+5)+7(2x+1)$.
$3x+15+14x+7$
$17x+22$
You try
Simplify $5(2x-3)-7(5x-1)$.
Pen and paper out — try it before you reveal.
$10x-15-35x+7$
$-25x-8$
$-25x-8$
You try
Simplify $p(3p-1)-7(3p-1)$.
Pen and paper out — try it before you reveal.
$3p^2-p-21p+7$
$3p^2-22p+7$
$3p^2-22p+7$
Section 2 of 5
More Double Brackets
Worked example — with the grid
Simplify $(x+5)(x+7)$.
$x(x+7)+5(x+7)$
$x^2+7x+5x+35$
$x^2+12x+35$
Worked example
Simplify $(2x+3)(6x+8)$.
$2x(6x+8)+3(6x+8)$
$12x^2+16x+18x+24$
$12x^2+34x+24$
Worked example
Simplify $(2x+5)(3x+9)$.
$2x(3x+9)+5(3x+9)$
$6x^2+18x+15x+45$
$6x^2+33x+45$
You try
Simplify $(7x-5)(2x-9)$.
Pen and paper out — try it before you reveal.
$7x(2x-9)-5(2x-9)$
$14x^2-63x-10x+45$
$14x^2-73x+45$
$14x^2-73x+45$
Section 3 of 5
Perfect Squares
$(x+5)^2$ means $(x+5)(x+5)$ — always write it out twice, then expand. Remember $7^2 = 7\times 7 = 49$.
Worked example
Expand $(x+5)^2$.
$(x+5)(x+5) = x(x+5)+5(x+5)$
$x^2+5x+5x+25$
$x^2+10x+25$
Worked example
Expand $(x+8)^2$.
$(x+8)(x+8) = x^2+8x+8x+64$
$x^2+16x+64$
Worked example
Expand $(2x-3)^2$.
$(2x-3)(2x-3) = 2x(2x-3)-3(2x-3)$
$4x^2-6x-6x+9$
$4x^2-12x+9$
Worked example
Expand $(5y+3)^2$.
$(5y+3)(5y+3) = 25y^2+15y+15y+9$
$25y^2+30y+9$
You try
Expand $(5x-9)^2$.
Pen and paper out — try it before you reveal.
$(5x-9)(5x-9) = 5x(5x-9)-9(5x-9)$
$25x^2-45x-45x+81$
$25x^2-90x+81$
$25x^2-90x+81$
You try
Expand $(6p-7)^2$.
Pen and paper out — try it before you reveal.
$(6p-7)(6p-7) = 6p(6p-7)-7(6p-7)$
$36p^2-42p-42p+49$
$36p^2-84p+49$
$36p^2-84p+49$
You try
Expand $(2x-7)^2$.
Pen and paper out — try it before you reveal.
$(2x-7)(2x-7) = 2x(2x-7)-7(2x-7)$
$4x^2-14x-14x+49$
$4x^2-28x+49$
$4x^2-28x+49$
Section 4 of 5
Longer Expressions
When one bracket has three terms, split it the same way — multiply the whole trinomial by each term of the first bracket.
Worked example
Simplify $(x+2)(x^2+3x+5)$.
$x(x^2+3x+5)+2(x^2+3x+5)$
$x^3+3x^2+5x+2x^2+6x+10$
$x^3+5x^2+11x+10$
Worked example
Simplify $(2p-1)(p^2-3p+5)$.
$2p(p^2-3p+5)-1(p^2-3p+5)$
$2p^3-6p^2+10p-p^2+3p-5$
$2p^3-7p^2+13p-5$
You try
Simplify $(3a-5)(a^2-6a-8)$.
Pen and paper out — try it before you reveal.
$3a(a^2-6a-8)-5(a^2-6a-8)$
$3a^3-18a^2-24a-5a^2+30a+40$
$3a^3-23a^2+6a+40$
$3a^3-23a^2+6a+40$
Section 5 of 5
The Array Method
The array (box) method lays every term in a grid. Fill each cell, then add the like terms inside.
Worked example
Simplify $(x+3)(x+2)$ using the array.
$x^2+2x+3x+6 = x^2+5x+6$
Worked example
Simplify $(2x+1)(3x+2)$.
$6x^2+3x+4x+2 = 6x^2+7x+2$
Worked example
Simplify $(5x-2)(3x-1)$.
$15x^2-6x-5x+2 = 15x^2-11x+2$
You try
Use the array to simplify $(2a-3)(a^2+5a+6)$.
Pen and paper out — try it before you reveal.
Grid columns $a^2,\,5a,\,6$ and rows $2a,\,-3$:
$2a^3+10a^2+12a-3a^2-15a-18$
$2a^3+7a^2-3a-18$
$2a^3+7a^2-3a-18$
That’s squares & arrays.
Perfect squares written out twice, trinomials split term by term, and the array grid for anything.