Factoring Algebraic Expressions Using Common Factors and Identities

Hello. In the previous lesson, you expanded brackets and established three key identities: the squares of a sum and difference, and the product of conjugates. Factorisation reverses that process: instead of multiplying factors to produce an expression, you recognise an expression’s structure and write it as a product.

This is a high-return skill for entrance exams. A factored form often exposes cancellation, makes substitution quicker, and later lets you solve equations efficiently. In this lesson, you will learn a disciplined sequence: first extract any common factor, then look for an exact standard-identity pattern, and finally check whether further factorisation is possible.

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Factorisation is reverse distribution

To factorise an expression means to rewrite it as multiplication. For example,

$$6x^2+18x$$

has $6x$ in both terms. Since

$$6x(x+3)=6x^2+18x,$$

we can factorise it as

$$\boxed{6x^2+18x=6x(x+3)}$$

The number or expression placed outside the bracket is a common factor: it divides every term exactly.

A useful way to avoid mistakes is to divide each original term by the factor you are taking out:

$$12x^3-20x^2$$

The greatest common factor of the coefficients $12$ and $20$ is $4$. Both terms also contain $x^2$, so the greatest common factor is $4x^2$.

$$12x^3 \div 4x^2=3x$$ $$-20x^2 \div 4x^2=-5$$

Therefore,

$$\boxed{12x^3-20x^2=4x^2(3x-5)}$$

The signs inside the bracket come directly from division. Never change a sign merely because it “looks more natural.”

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Finding the greatest common factor

For a polynomial with several terms, construct the GCF from two parts:

1. The greatest common numerical divisor of all coefficients.
2. Every variable common to all terms, taken to its lowest exponent.

Consider:

$$18x^3y^2-24x^2y+6xy^3$$

The coefficient GCF is $6$. For the variables:

• the lowest power of $x$ is $x^1$;
• the lowest power of $y$ is $y^1$.

So the GCF is $6xy$:

$$18x^3y^2-24x^2y+6xy^3$$ $$=6xy(3x^2y-4x+y^2)$$

Check it by multiplying back:

$$6xy(3x^2y-4x+y^2)$$ $$=18x^3y^2-24x^2y+6xy^3$$

A variable missing from even one term cannot belong to the common factor. For example, in

$$8x^2+12x+4,$$

the GCF is $4$, not $4x$, because the final term has no $x$.

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A repeated bracket can also be a common factor

The common factor need not be a number or a single variable. Sometimes an entire bracket repeats:

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

Both terms contain $3x+1$. Treat that bracket as one object:

$$\boxed{x(3x+1)-5(3x+1)=(3x+1)(x-5)}$$

This is still reverse distribution:

$$(3x+1)(x-5)=x(3x+1)-5(3x+1)$$

The key visual habit is to scan for a repeated complete bracket before doing any expansion.

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Factor out the GCF first

A standard exam rule is:

$$\boxed{\text{Extract the GCF before attempting an identity.}}$$

Why? A hidden common factor can conceal an identity.

For example:

$$8x^2+24x+18$$

All terms are divisible by $2$:

$$8x^2+24x+18=2(4x^2+12x+9)$$

Now the bracket is a perfect-square trinomial:

$$4x^2=(2x)^2$$ $$9=3^2$$ $$12x=2(2x)(3)$$

Therefore,

$$\boxed{8x^2+24x+18=2(2x+3)^2}$$

Stopping at $2(4x^2+12x+9)$ is partially correct, but not fully factorised.

Sometimes extracting a negative factor makes the remaining structure clearer:

$$-8x^2+18$$ $$=-2(4x^2-9)$$

The bracket is now a difference of squares:

$$4x^2-9=(2x)^2-3^2$$

So,

$$\boxed{-8x^2+18=-2(2x+3)(2x-3)}$$

Taking out a negative GCF is not compulsory, but it is often the cleanest choice when the leading term is negative.

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Perfect-square trinomials

From the previous lesson:

$$(a+b)^2=a^2+2ab+b^2$$ $$(a-b)^2=a^2-2ab+b^2$$

Reversing these identities gives the factorisation forms:

$$\boxed{a^2+2ab+b^2=(a+b)^2}$$ $$\boxed{a^2-2ab+b^2=(a-b)^2}$$

A perfect-square trinomial must meet all three conditions:

• the first term is a perfect square;
• the last term is a perfect square;
• the middle term is exactly twice the product of their square roots.

Consider:

$$25p^2-30pq+9q^2$$

The first and last terms are squares:

$$25p^2=(5p)^2$$ $$9q^2=(3q)^2$$

Now test the middle term:

$$-2(5p)(3q)=-30pq$$

It matches exactly, so:

$$\boxed{25p^2-30pq+9q^2=(5p-3q)^2}$$

The sign of the middle term decides the sign inside the bracket:

Expression pattern	Factorised form
$a^2+2ab+b^2$	$(a+b)^2$
$a^2-2ab+b^2$	$(a-b)^2$

Do not factor an expression as a perfect square merely because its first and last terms are squares. For instance,

$$x^2+6x+16$$

has square end terms, but the relevant middle term for $(x+4)^2$ would be

$$2(x)(4)=8x,$$

not $6x$. So this is not a perfect-square trinomial.

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Difference of two squares

The third essential identity from the previous lesson was:

$$(a+b)(a-b)=a^2-b^2$$

Reversing it gives:

$$\boxed{a^2-b^2=(a+b)(a-b)}$$

Use this identity only when all of the following are true:

1. There are exactly two main terms.
2. The terms are separated by subtraction.
3. Both terms are perfect squares.

For example:

$$9y^4-25x^2$$

can be written as

$$(3y^2)^2-(5x)^2$$

Therefore,

$$\boxed{9y^4-25x^2=(3y^2+5x)(3y^2-5x)}$$

The two factors are conjugates: the terms are the same, but the internal signs differ.

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The image also explains why this identity is geometrically sound: the leftover L-shaped region from the larger square can be rearranged into a rectangle. Its area has not changed, only its shape has.

Be alert to what the identity does not say:

$$a^2+b^2$$

is a sum of squares, not a difference of squares. It does not factor using this identity over integers.

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Combining methods: factor completely

Entrance-exam expressions are often designed in layers. After extracting a GCF, inspect the remaining bracket again.

Consider:

$$12x^3-3x$$

First take out the GCF $3x$:

$$12x^3-3x=3x(4x^2-1)$$

The bracket is a difference of squares:

$$4x^2-1=(2x)^2-1^2$$

Factor it again:

$$\boxed{12x^3-3x=3x(2x+1)(2x-1)}$$

This is the fully factorised form.

The following recognition sequence is reliable under time pressure:

1. Count terms and scan for a GCF. Look for common numbers, variables, or complete brackets.
2. Extract the GCF. Divide every term carefully.
3. Inspect what remains. Check for a perfect-square trinomial or a difference of squares.
4. Repeat if necessary. A factored bracket may itself factor further.
5. Verify by expansion. This is especially useful when signs or powers are involved.

For the final check of the previous example:

$$(2x+1)(2x-1)=4x^2-1$$

and then

$$3x(4x^2-1)=12x^3-3x$$

The original expression is recovered.

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Key takeaways

Factorisation is reverse expansion: you rewrite a sum or difference as a product.

Start with the GCF. For monomials, it consists of the numerical GCF and the lowest shared exponent of each common variable. A complete bracket can also be a common factor.

The three core identity patterns are:

$$a^2+2ab+b^2=(a+b)^2$$ $$a^2-2ab+b^2=(a-b)^2$$ $$a^2-b^2=(a+b)(a-b)$$

The most important habit is structural recognition: do not force an identity because an expression merely resembles one. Confirm the exact middle term in a perfect-square trinomial, and confirm two perfect squares separated by subtraction for a difference of squares.

Next, you will use algebraic manipulation in a more operational setting: solving one-variable linear equations with integer and fractional coefficients.