Factoring polynomials: a step-by-step guide

Q: What is factored form?

A polynomial written as a product of factors, such as (x + 3)(x + 2) instead of x^2 + 5x + 6. Setting each factor to zero gives the roots.

Q: What does factoring completely mean?

Factoring until none of the factors can be factored again. 2x^3 - 18x = 2x(x^2 - 9) is not complete; 2x(x + 3)(x - 3) is.

The steps for factoring polynomials — common factor first, then two, three or four terms — with worked examples, each checked by multiplying back out.

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Factoring polynomials means writing a polynomial as a product of simpler polynomials — the reverse of expanding brackets. Expanding $(x + 2)(x + 3)$ gives $x^2 + 5x + 6$ ; factoring $x^2 + 5x + 6$ takes you back to $(x + 2)(x + 3)$ . The product is called the factored form.

Factored form is useful: it shows where a polynomial equals zero (each factor gives a root), it simplifies fractions, and it makes graphs easier to sketch. In the UK the same skill is called factorising expressions.

Factoring steps: a checklist

Take out the greatest common factor (GCF) of all the terms.
Count the terms that are left.
Two terms: look for a difference of two squares, $a^2 - b^2 = (a + b)(a - b)$ , or a sum or difference of two cubes.
Three terms (a trinomial): find two numbers that multiply to $a \times c$ and add to $b$ .
Four terms: try grouping them in pairs.
Check each factor — can it be factored again? Then multiply out to check.

Common monomial factor: examples with answers

A common monomial factor is a single term that divides every term of the polynomial. Find the GCF of the numbers and the lowest power of each letter:

$6x^2 + 9x = 3x(2x + 3)$
$4x^3 - 8x^2 = 4x^2(x - 2)$
$5a^2b + 10ab^2 = 5ab(a + 2b)$
$7y - 21 = 7(y - 3)$

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Easy: take out the greatest common factor

Factor $12x^3 + 18x^2$ .

Onyesha hatua za suluhisho

The GCF of 12 and 18 is 6, and the lowest power of $x$ is $x^2$ , so the GCF is $6x^2$ . $12x^3 + 18x^2 = 6x^2 \cdot 2x + 6x^2 \cdot 3$
Divide each term by $6x^2$ . $6x^2(2x + 3)$
Check by expanding. $6x^2 \cdot 2x + 6x^2 \cdot 3 = 12x^3 + 18x^2 \checkmark$

Jibu: $6x^2(2x + 3)$

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Medium: a trinomial with leading coefficient 1

Factor $x^2 + 5x + 6$ .

Onyesha hatua za suluhisho

Find two numbers that multiply to 6 and add to 5: 2 and 3. $2 \times 3 = 6,\quad 2 + 3 = 5$
Write the factors. $(x + 3)(x + 2)$
Check. $(x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6 \checkmark$

Jibu: $(x + 3)(x + 2)$

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Medium: the difference of two squares

Factor $9x^2 - 25$ .

Onyesha hatua za suluhisho

Both terms are perfect squares: $9x^2 = (3x)^2$ and $25 = 5^2$ . $a = 3x,\quad b = 5$
Use $a^2 - b^2 = (a + b)(a - b)$ . $(3x + 5)(3x - 5)$

Jibu: $(3x + 5)(3x - 5)$

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Hard: a trinomial with leading coefficient other than 1

Factor $6x^2 + 7x - 3$ .

Onyesha hatua za suluhisho

Multiply $a \times c = 6 \times (-3) = -18$ . Two numbers that multiply to $-18$ and add to 7 are 9 and $-2$ . $9 \times (-2) = -18,\quad 9 + (-2) = 7$
Split the middle term and group. $6x^2 + 9x - 2x - 3 = 3x(2x + 3) - 1(2x + 3)$
Take out the common bracket. $(2x + 3)(3x - 1)$
Check. $6x^2 - 2x + 9x - 3 = 6x^2 + 7x - 3 \checkmark$

Jibu: $(2x + 3)(3x - 1)$

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Hard: four terms, factored by grouping

Factor $x^3 + 2x^2 - 9x - 18$ .

Onyesha hatua za suluhisho

Group in pairs and take out a factor from each. $x^2(x + 2) - 9(x + 2)$
Take out the common bracket $(x + 2)$ . $(x + 2)(x^2 - 9)$
$x^2 - 9$ is a difference of squares, so factor it too. $(x + 2)(x + 3)(x - 3)$

Jibu: $(x + 3)(x + 2)(x - 3)$

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Factoring completely

Factoring completely means continuing until no factor can be broken down any further. The most common slip is stopping after the first step. In the example below, taking out $2x$ leaves $x^2 - 9$ , which still factors.

Mfano uliotatuliwa

Factor this polynomial completely

Factor $2x^3 - 18x$ completely.

Onyesha hatua za suluhisho

Take out the GCF, $2x$ . $2x(x^2 - 9)$
Factor the difference of squares. $2x(x + 3)(x - 3)$

Jibu: $2x(x + 3)(x - 3)$

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Special patterns to recognise

Perfect square trinomial: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$ . Example: $x^2 - 10x + 25 = (x - 5)^2$ .
Difference of two squares: $a^2 - b^2 = (a + b)(a - b)$ .
Difference of two cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .
Sum of two cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .

Mfano uliotatuliwa

A perfect square trinomial

Factor $x^2 + 6x + 9$ .

Onyesha hatua za suluhisho

The first and last terms are squares ( $x^2$ and $3^2$ ), and the middle term is $2 \times x \times 3$ . $x^2 + 2(x)(3) + 3^2$
So it is a perfect square. $(x + 3)^2$

Jibu: $(x + 3)^2$

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A difference of two cubes

Factor $x^3 - 8$ .

Onyesha hatua za suluhisho

Both terms are cubes: $x^3$ and $8 = 2^3$ , so $a = x$ and $b = 2$ . $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
Substitute. $(x - 2)(x^2 + 2x + 4)$
The quadratic factor has no real roots, so this is fully factored. $\checkmark$

Jibu: $(x - 2)(x^2 + 2x + 4)$

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Why factored form is useful

Factored form turns hard equations into easy ones. To solve $x^2 + 5x + 6 = 0$ , factor it as $(x + 3)(x + 2) = 0$ . A product is zero only when a factor is zero, so $x = -3$ or $x = -2$ . The same idea finds where a graph crosses the x-axis and simplifies algebraic fractions such as $\frac{x^2 - 9}{x + 3} = x - 3$ (for $x \neq -3$ ).

Common mistakes

Skipping the common factor. Without it, $2x^3 - 18x$ looks much harder than it is.
Sign errors in the two numbers: for $6x^2 + 7x - 3$ the product is $-18$ , so one number must be negative.
Stopping too early — always check whether each factor can be factored again.
Treating a sum of squares like a difference: $x^2 + 25$ does not factor as $(x + 5)(x - 5)$ , which expands to $x^2 - 25$ .

Maswali yanayoulizwa mara kwa mara

What is factored form?

A polynomial written as a product of factors, such as $(x + 3)(x + 2)$ instead of $x^2 + 5x + 6$ . Setting each factor to zero gives the roots.

What are the steps for factoring polynomials?

Take out the greatest common factor, then count the terms: two terms — difference of squares; three terms — find two numbers for the trinomial; four terms — grouping. Repeat on each factor, then check by expanding.

What does factoring completely mean?

Factoring until none of the factors can be factored again. $2x^3 - 18x = 2x(x^2 - 9)$ is not complete; $2x(x + 3)(x - 3)$ is.

How do I factorise expressions with a common factor?

Find the biggest number that divides every term and the lowest power of each letter that appears in every term, write them outside a bracket as one factor, and divide each term by that factor to fill the bracket: $10x^2 + 15x = 5x(2x + 3)$ .

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Factoring polynomials: a step-by-step guide

Factoring steps: a checklist

Common monomial factor: examples with answers

Easy: take out the greatest common factor

Medium: a trinomial with leading coefficient 1

Medium: the difference of two squares

Hard: a trinomial with leading coefficient other than 1

Hard: four terms, factored by grouping

Factoring completely

Factor this polynomial completely

Special patterns to recognise

A perfect square trinomial

A difference of two cubes

Why factored form is useful

Common mistakes

Maswali yanayoulizwa mara kwa mara

What is factored form?

What are the steps for factoring polynomials?

What does factoring completely mean?

How do I factorise expressions with a common factor?

Endelea kujifunza

Vikokotoo

Mazoezi