Algebra · Junior Cert
Factorising
Junior Cert Higher · The Guide Number method · Tap NEXT to begin
Section 1 of 6
Common Factor
Factorising is expanding in reverse: take the highest common factor outside a bracket.
Worked example — take out the HCF
Factorise $6x + 9$.
HCF of $6$ and $9$ is $3$
$3(2x+3)$
Worked example — a factor with a letter
Factorise $4x^2 + 8x$.
HCF is $4x$
$4x(x+2)$
Worked example — a bigger one
Factorise $12x^2 + 8x$.
HCF is $4x$
$4x(3x+2)$
You try
Factorise $10x + 15$.
HCF of $10$ and $15$ is $5$.
$5(2x + 3)$
$5(2x+3)$
You try
Factorise $6x^2 - 9x$.
HCF is $3x$.
$3x(2x - 3)$
$3x(2x-3)$
Section 2 of 6
Difference of Two Squares
The pattern
$a^2 - b^2 = (a-b)(a+b)$ — spot it when there is no middle term and a minus sign.
Worked example
Factorise $x^2 - 49$.
No middle term and a minus → difference of two squares
Write as $(x)^2 - 7^2$
$(x - 7)(x + 7)$
Worked example
Factorise $4x^2 - 25$.
No middle term and a minus → difference of two squares
Write as $(2x)^2 - 5^2$
$(2x - 5)(2x + 5)$
Worked example
Factorise $9x^2 - 49$.
No middle term and a minus → difference of two squares
Write as $(3x)^2 - 7^2$
$(3x - 7)(3x + 7)$
You try
Factorise $x^2 - 81$.
It’s $a^2-b^2=(a-b)(a+b)$. Here $a=x,\ b=9$.
No middle term and a minus → difference of two squares
Write as $(x)^2 - 9^2$
$(x - 9)(x + 9)$
$(x - 9)(x + 9)$
You try
Factorise $25x^2 - 16$.
It’s $a^2-b^2=(a-b)(a+b)$. Here $a=5x,\ b=4$.
No middle term and a minus → difference of two squares
Write as $(5x)^2 - 4^2$
$(5x - 4)(5x + 4)$
$(5x - 4)(5x + 4)$
You try
Factorise $4x^2 - 81$.
It’s $a^2-b^2=(a-b)(a+b)$. Here $a=2x,\ b=9$.
No middle term and a minus → difference of two squares
Write as $(2x)^2 - 9^2$
$(2x - 9)(2x + 9)$
$(2x - 9)(2x + 9)$
Section 3 of 6
Guide Number: Positive Constant
First see where a trinomial comes from. Expand $(x+3)(x+2)$: $x(x+2)+3(x+2) = x^2 + 5x + 6$. Factorising reverses this.
★ The Guide Number method
1. Guide Number GN = a × c (first number times the last, keep the sign).
2. List the factor pairs of the GN.
3. Pick the pair that gives b (the middle): Add if GN is +, Sub if GN is −.
4. Split the middle, then group and take out the common bracket.
2. List the factor pairs of the GN.
3. Pick the pair that gives b (the middle): Add if GN is +, Sub if GN is −.
4. Split the middle, then group and take out the common bracket.
When the last number is positive, the two numbers Add to the middle. Here $a=1$.
Worked example
Factorise $x^2 + 5x + 6$.
GN $= 1\times6 = 6$ (Add)
Factor pairs of $6$: $1\times6$, $2\times3$
The pair that adds to $5$: $3$ and $2$
Split: $x^2 + 3x + 2x + 6$
Group: $x(x + 3) + 2(x + 3)$
$(x + 2)(x + 3)$
Worked example
Factorise $x^2 + 10x + 21$.
GN $= 1\times21 = 21$ (Add)
Factor pairs of $21$: $1\times21$, $3\times7$
The pair that adds to $10$: $7$ and $3$
Split: $x^2 + 7x + 3x + 21$
Group: $x(x + 7) + 3(x + 7)$
$(x + 3)(x + 7)$
Worked example
Factorise $x^2 + 15x + 50$.
GN $= 1\times50 = 50$ (Add)
Factor pairs of $50$: $1\times50$, $2\times25$, $5\times10$
The pair that adds to $15$: $10$ and $5$
Split: $x^2 + 10x + 5x + 50$
Group: $x(x + 10) + 5(x + 10)$
$(x + 5)(x + 10)$
Section 4 of 6
Guide Number: Negative Constant
When the last number is negative, the GN is negative — now the pair Sub (one plus, one minus) to give the middle.
Worked example
Factorise $x^2 - 2x - 8$.
GN $= 1\times(-8) = -8$ (Sub)
Factor pairs of $8$: $1\times8$, $2\times4$
The pair that subtracts to $-2$: $-4$ and $2$
Split: $x^2 - 4x + 2x - 8$
Group: $x(x - 4) + 2(x - 4)$
$(x + 2)(x - 4)$
Worked example
Factorise $x^2 - 2x - 24$.
GN $= 1\times(-24) = -24$ (Sub)
Factor pairs of $24$: $1\times24$, $2\times12$, $3\times8$, $4\times6$
The pair that subtracts to $-2$: $-6$ and $4$
Split: $x^2 - 6x + 4x - 24$
Group: $x(x - 6) + 4(x - 6)$
$(x + 4)(x - 6)$
Worked example
Factorise $x^2 + 2x - 15$.
GN $= 1\times(-15) = -15$ (Sub)
Factor pairs of $15$: $1\times15$, $3\times5$
The pair that subtracts to $2$: $-3$ and $5$
Split: $x^2 - 3x + 5x - 15$
Group: $x(x - 3) + 5(x - 3)$
$(x + 5)(x - 3)$
Section 5 of 6
Guide Number: With a Coefficient
Now a number in front of $x^2$. Same method — the GN $= a\times c$ just gets bigger. Positive constant first, then negative.
Worked example
Factorise $3x^2 + 17x + 10$.
GN $= 3\times10 = 30$ (Add)
Factor pairs of $30$: $1\times30$, $2\times15$, $3\times10$, $5\times6$
The pair that adds to $17$: $15$ and $2$
Split: $3x^2 + 15x + 2x + 10$
Group: $3x(x + 5) + 2(x + 5)$
$(3x + 2)(x + 5)$
Worked example
Factorise $2x^2 + 7x + 3$.
GN $= 2\times3 = 6$ (Add)
Factor pairs of $6$: $1\times6$, $2\times3$
The pair that adds to $7$: $6$ and $1$
Split: $2x^2 + 6x + x + 3$
Group: $2x(x + 3) + 1(x + 3)$
$(2x + 1)(x + 3)$
Worked example
Factorise $3x^2 - 5x - 12$.
GN $= 3\times(-12) = -36$ (Sub)
Factor pairs of $36$: $1\times36$, $2\times18$, $3\times12$, $4\times9$, $6\times6$
The pair that subtracts to $-5$: $-9$ and $4$
Split: $3x^2 - 9x + 4x - 12$
Group: $3x(x - 3) + 4(x - 3)$
$(3x + 4)(x - 3)$
Worked example
Factorise $2x^2 - x - 6$.
GN $= 2\times(-6) = -12$ (Sub)
Factor pairs of $12$: $1\times12$, $2\times6$, $3\times4$
The pair that subtracts to $-1$: $-4$ and $3$
Split: $2x^2 - 4x + 3x - 6$
Group: $2x(x - 2) + 3(x - 2)$
$(2x + 3)(x - 2)$
Section 6 of 6
Mixed — Try Yourself
All types mixed. Work out the GN, decide Add or Sub, list the factor pairs, split and group.
You try
Factorise $x^2 - 12x + 20$.
GN $= 1\times20 = 20$ (Add). Factor pairs: $1\times20$, $2\times10$, $4\times5$.
GN $= 1\times20 = 20$ (Add)
Factor pairs of $20$: $1\times20$, $2\times10$, $4\times5$
The pair that adds to $-12$: $-10$ and $-2$
Split: $x^2 - 10x - 2x + 20$
Group: $x(x - 10) - 2(x - 10)$
$(x - 2)(x - 10)$
$(x - 2)(x - 10)$
You try
Factorise $x^2 + 7x + 12$.
GN $= 1\times12 = 12$ (Add). Factor pairs: $1\times12$, $2\times6$, $3\times4$.
GN $= 1\times12 = 12$ (Add)
Factor pairs of $12$: $1\times12$, $2\times6$, $3\times4$
The pair that adds to $7$: $4$ and $3$
Split: $x^2 + 4x + 3x + 12$
Group: $x(x + 4) + 3(x + 4)$
$(x + 3)(x + 4)$
$(x + 3)(x + 4)$
You try
Factorise $x^2 - x - 12$.
GN $= 1\times(-12) = -12$ (Sub). Factor pairs: $1\times12$, $2\times6$, $3\times4$.
GN $= 1\times(-12) = -12$ (Sub)
Factor pairs of $12$: $1\times12$, $2\times6$, $3\times4$
The pair that subtracts to $-1$: $3$ and $-4$
Split: $x^2 + 3x - 4x - 12$
Group: $x(x + 3) - 4(x + 3)$
$(x - 4)(x + 3)$
$(x - 4)(x + 3)$
You try
Factorise $x^2 + 3x - 10$.
GN $= 1\times(-10) = -10$ (Sub). Factor pairs: $1\times10$, $2\times5$.
GN $= 1\times(-10) = -10$ (Sub)
Factor pairs of $10$: $1\times10$, $2\times5$
The pair that subtracts to $3$: $-2$ and $5$
Split: $x^2 - 2x + 5x - 10$
Group: $x(x - 2) + 5(x - 2)$
$(x + 5)(x - 2)$
$(x + 5)(x - 2)$
You try
Factorise $x^2 + 8x + 15$.
GN $= 1\times15 = 15$ (Add). Factor pairs: $1\times15$, $3\times5$.
GN $= 1\times15 = 15$ (Add)
Factor pairs of $15$: $1\times15$, $3\times5$
The pair that adds to $8$: $5$ and $3$
Split: $x^2 + 5x + 3x + 15$
Group: $x(x + 5) + 3(x + 5)$
$(x + 3)(x + 5)$
$(x + 3)(x + 5)$
You try
Factorise $5x^2 - 17x + 6$.
GN $= 5\times6 = 30$ (Add). Factor pairs: $1\times30$, $2\times15$, $3\times10$, $5\times6$.
GN $= 5\times6 = 30$ (Add)
Factor pairs of $30$: $1\times30$, $2\times15$, $3\times10$, $5\times6$
The pair that adds to $-17$: $-15$ and $-2$
Split: $5x^2 - 15x - 2x + 6$
Group: $5x(x - 3) - 2(x - 3)$
$(5x - 2)(x - 3)$
$(5x - 2)(x - 3)$
You try
Factorise $3x^2 - 2x - 8$.
GN $= 3\times(-8) = -24$ (Sub). Factor pairs: $1\times24$, $2\times12$, $3\times8$, $4\times6$.
GN $= 3\times(-8) = -24$ (Sub)
Factor pairs of $24$: $1\times24$, $2\times12$, $3\times8$, $4\times6$
The pair that subtracts to $-2$: $-6$ and $4$
Split: $3x^2 - 6x + 4x - 8$
Group: $3x(x - 2) + 4(x - 2)$
$(3x + 4)(x - 2)$
$(3x + 4)(x - 2)$
You try
Factorise $2x^2 + 5x - 3$.
GN $= 2\times(-3) = -6$ (Sub). Factor pairs: $1\times6$, $2\times3$.
GN $= 2\times(-3) = -6$ (Sub)
Factor pairs of $6$: $1\times6$, $2\times3$
The pair that subtracts to $5$: $6$ and $-1$
Split: $2x^2 + 6x - x - 3$
Group: $2x(x + 3) - 1(x + 3)$
$(2x - 1)(x + 3)$
$(2x - 1)(x + 3)$
You try
Factorise $3x^2 + 10x + 8$.
GN $= 3\times8 = 24$ (Add). Factor pairs: $1\times24$, $2\times12$, $3\times8$, $4\times6$.
GN $= 3\times8 = 24$ (Add)
Factor pairs of $24$: $1\times24$, $2\times12$, $3\times8$, $4\times6$
The pair that adds to $10$: $6$ and $4$
Split: $3x^2 + 6x + 4x + 8$
Group: $3x(x + 2) + 4(x + 2)$
$(3x + 4)(x + 2)$
$(3x + 4)(x + 2)$
You try
Factorise $2x^2 - 7x + 3$.
GN $= 2\times3 = 6$ (Add). Factor pairs: $1\times6$, $2\times3$.
GN $= 2\times3 = 6$ (Add)
Factor pairs of $6$: $1\times6$, $2\times3$
The pair that adds to $-7$: $-6$ and $-1$
Split: $2x^2 - 6x - x + 3$
Group: $2x(x - 3) - 1(x - 3)$
$(2x - 1)(x - 3)$
$(2x - 1)(x - 3)$
That’s Factorising.
Common factor, difference of two squares, and the Guide Number — Add when it’s positive, Sub when it’s negative, even with a number in front.