# Factorising

## Factorising Revision

**Factorising into Single Brackets**

**Factorising** is putting expressions into brackets, this is the reverse of expanding brackets.

Make sure you are happy with the following topics before continuing.

**Factorising into single brackets – 3 Key steps**

**Example:**

Fully factorise the following:

\textcolor{red}{12}\textcolor{limegreen}{x^2} +\textcolor{red}{8}\textcolor{limegreen}{x}

**Step 1 **– Take out the largest common factor of both the **numbers, **and place it in front of the brackets.

Factors of \textcolor{red}{12} are 1, 2, 3, \textcolor{blue}{4}, 6, 12

Factors of \textcolor{red}{8} are 1, 2, \textcolor{blue}{4}, 8

The largest common factor is \bf{\textcolor{blue}{4}}

\textcolor{red}{12} = \textcolor{blue}{4} \times \textcolor{purple}{3}

\textcolor{red}{8} = \textcolor{blue}{4} \times \textcolor{purple}{2}

**Step 2 –** Take out the **highest power** of the “Letter” which is a part of every term.

\begin{aligned}\textcolor{red}{12}\textcolor{limegreen}{x^2} = & \textcolor{red}{12} \times \textcolor{limegreen}{x} \times \xcancel{\textcolor{limegreen}{x}}\\ \textcolor{red}{8}\textcolor{limegreen}{x} = & \textcolor{red}{8} \times \xcancel{\textcolor{limegreen}{x}}\end{aligned}

We can take out one \textcolor{limegreen}{x} from each term, placing in front of the brackets.

**Step 3 – **Place the highest factor and highest power of the letter in front of the bracket, then add the remaining terms inside the bracket

\textcolor{blue}{4}\textcolor{limegreen}{x}(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\textcolor{blue}{4}\textcolor{limegreen}{x}(\textcolor{purple}{3}\textcolor{limegreen}{x} +\textcolor{purple}{2})

To check, you can multiply the bracket back out to see if you have the right answer.

4x(3x+2) = 12x^2 +8x

## Example 1:** Factorising Two Terms**

Fully Factorise the following, \textcolor{red}{3}\textcolor{limegreen}{x}\textcolor{Orange}{y} + \textcolor{red}{6}\textcolor{limegreen}{x^2}.

**[2 marks]**

**Step 1 –** Take out the largest **common factor** of both the numbers, and place it in front of the brackets.

Factors of \textcolor{red}{3} are 1, \textcolor{blue}{3}

Factors of \textcolor{red}{6} are 1, 2, \textcolor{blue}{3}, 6

The largest common factor is \bf{\textcolor{blue}{3}}

\textcolor{red}{3} = \textcolor{blue}{3} \times \textcolor{purple}{1}

\textcolor{red}{6} = \textcolor{blue}{3} \times \textcolor{purple}{2}

**Step 2 –** Take out the highest power of the “Letter” which is a part of every term.

\begin{aligned}\textcolor{red}{3}\textcolor{limegreen}{x}\textcolor{Orange}{y} = & \textcolor{red}{3} \times \xcancel{\textcolor{limegreen}{x}} \times \textcolor{orange}{y}\\ \textcolor{red}{6}\textcolor{limegreen}{x^2} = & \textcolor{red}{6} \times \textcolor{limegreen}{x} \times \xcancel{\textcolor{limegreen}{x}}\end{aligned}

We can take out one \textcolor{limegreen}{x} from each term, placing it in front of the brackets.

**Step 3 –** Place the highest factor and highest letter in front of the bracket, then add the remaining terms inside the bracket

\textcolor{blue}{3}\textcolor{limegreen}{x}(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\textcolor{blue}{3}\textcolor{limegreen}{x}(\textcolor{orange}{y} +\textcolor{purple}{2}\textcolor{limegreen}{x})

To check you can multiply the bracket back out to see if you have the right answer.

3x(y+2x) = 3xy+6x^2

**Example 2: Factorising with three terms**

Fully factorise the following, \textcolor{red}{8}\textcolor{limegreen}{x}\textcolor{Orange}{y} + \textcolor{red}{12}\textcolor{limegreen}{x^2}\textcolor{Orange}{y} - \textcolor{red}{4}\textcolor{limegreen}{x^2}\textcolor{Orange}{y^2}.

**[3 marks]**

**Step 1 –** Take out the largest common factor of all of the numbers, and place it in front of the brackets.

Factors of \textcolor{red}{8} are 1, 2, \textcolor{blue}{4}, 8

Factors of \textcolor{red}{12} are 1, 2, 3, \textcolor{blue}{4}, 6, 12

Factors of \textcolor{red}{4} are 1, 2, \textcolor{blue}{4}

The highest common factor of all three is \textcolor{blue}{4}

\textcolor{red}{8} = \textcolor{blue}{4} \times \textcolor{purple}{2}

\textcolor{red}{12} = \textcolor{blue}{4} \times \textcolor{purple}{3}

\textcolor{red}{4}= \textcolor{blue}{4} \times \textcolor{purple}{1}

**Step 2 –** Take out the highest power of the “Letter” which is a part of every term.

\begin{aligned}\textcolor{red}{8}\textcolor{limegreen}{x}\textcolor{Orange}{y} = & \textcolor{red}{8} \times \xcancel{\textcolor{limegreen}{x}} \times \xcancel{\textcolor{Orange}{y}}\\ \textcolor{red}{12}\textcolor{limegreen}{x^2}\textcolor{Orange}{y} = & \textcolor{red}{12} \times \xcancel{\textcolor{limegreen}{x}} \times\textcolor{limegreen}{x} \times \xcancel{\textcolor{Orange}{y}} \\ \textcolor{red}{4}\textcolor{limegreen}{x^2}\textcolor{Orange}{y^2} = & \textcolor{red}{4} \times \xcancel{\textcolor{limegreen}{x}} \times \textcolor{limegreen}{x} \times \xcancel{\textcolor{Orange}{y}} \times \textcolor{Orange}{y} \end{aligned}

We can take out one \textcolor{limegreen}{x} and one \textcolor{Orange}{y} from each term.

**Step 3 –** Place the highest factor and highest letter in front of the bracket, then add the remaining terms inside the bracket.

\textcolor{blue}{4}\textcolor{limegreen}{x}\textcolor{Orange}{y}(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\textcolor{blue}{4}\textcolor{limegreen}{x}\textcolor{Orange}{y}(\textcolor{purple}{2} + \textcolor{purple}{3}\textcolor{limegreen}{x} - \textcolor{limegreen}{x}\textcolor{Orange}{y})

To check you can multiply the bracket back out to see if you have the right answer.

4xy(2 + 3x-xy) = 8xy + 12x^2y-4x^2y^2

## Factorising Example Questions

**Question 1:** Factorise fully 10pq + 15pqr

**[2 marks]**

Take out a factor of 5 from both terms to get

5(2pq + 3pqr)

There is both a p and a q in the two terms inside the bracket. Taking out both p and q, we get

5pq(2 + 3r)

The two numbers in the bracket have nothing more in common so we are done.

**Question 2:** Factorise fully u^3+3uv^3+2u

**[2 marks]**

Take out a factor of u from both terms to get,

u(u^2+3v^3+2)

The terms inside the bracket have no more common factors, so we are done.

**Question 3:** Factorise fully 4xy^5 + y^5 + 12y^7

**[2 marks]**

The first and last term have a factor of 4 in common, but the middle term doesn’t, so we can’t take any numbers out as factors.

All 3 terms have a factor of y in them. Specifically, the highest power of y that all 3 terms have in common is y^5. Taking y^5 out as a factor, we get,

y^5(4x + 1 + 12y^2)

The terms in the bracket have no more common factors, so we are done.

**Question 4:** Factorise fully 5xy^2-5x^2y-5x^2y^2

**[3 marks]**

Take out a factor of 5 from every term to get

5(xy^2-x^2y-x^2y^2)

Now, clearly each term has a factor of x and y, so we just need to determine what the highest power of each factor we can take out is,

5xy(y-x-xy)

The terms inside the bracket have no more common factors, so we are done.

**Question 5:** Factorise fully 7abc + 14a^{2}bc + 21ab^{2}c + 49abc^3

**[3 marks]**

Take out a factor of 7 from every term to get

7(abc + 2a^{2}bc + 3ab^{2}c + 7abc^3)

Now, clearly each term has a factor of a, b, and c, so we just need to determine what the highest power of each factor we can take out is.

The first term only has the three factors a, b, and c to the power of 1 (note that we don’t write the power of 1 since x^1 = x), which means that this is the highest power of each factor we can take out is 1. Taking out a factor of a, b, and c, we get

7abc(1 + 2a + 3b + 7c^2)

The terms inside the bracket have no more common factors, so we are done.