## What you need to know

Expanding Single Brackets

We often come across mathematical expressions with brackets in them. Sometimes it’s useful to have them in brackets, but sometimes we’d rather not have them at all. The process by which we remove bracket is called expanding (or multiplying out) the brackets.

Note: this is the opposite process to factorising, which can you see more on here (https://mathsmadeeasy.co.uk/gcse-maths-revision/factorising-foundation-gcse-maths-revision-worksheets/) and here (https://mathsmadeeasy.co.uk/gcse-maths-revision/factorising-quadratics-gcse-maths-revision-worksheets/).

The first thing to understand about expanding brackets is this: is that if you ever see a number, algebraic term, or algebraic expression written directly before a bracket, such as $5(4+b)$, then this means that everything inside the bracket is being multiplied by what’s in front of it. So, when you are expanding a single bracket, make sure that whatever is out front gets multiplied by everything on the inside.

Example: Expand $2a(4 + a)$.

As stated, everything inside the bracket, a 4 and an $a$, is being multiplied what’s outside it, $2a$. So, the expansion of this bracket looks like

$2a(4 + a) = 2a \times 4 + 2a \times a = 8a + 2a^2$

Thus, the result of the expansion is $8a + 2a^2$.

The principle of expanding single brackets remains the same throughout, it is only ever made more complicated by introducing more algebraic terms. At this point it’s worth making sure you’re comfortable with the laws of indices (https://mathsmadeeasy.co.uk/gcse-maths-revision/rules-indices-gcse-maths-revision-worksheets/).

Example: Expand $5yx^2\left(3x^3 - 5xy + wy^2\right)$.

Just as before, we will multiply what’s outside the bracket, $5yx^{2}$, by everything inside the bracket. So, the bracket expansion looks like

\begin{aligned}5yx^2\left(3x^3+5xy+wy^2\right)&=5yx^2\times 3x^3+5yx^2\times (-5xy)+5yx^2\times wy^2 \\ &=15yx^5+25y^{2}x^3+5wy^{3}x^2\end{aligned}

Thus, the result of the expansion is $15yx^5+25y^{2}x^3+5wy^{3}x^2$.

Expanding Double Brackets

When expanding double brackets, the idea is similar: we need to multiply each of the things in the first bracket is by each of the things in the second bracket. One method of making sure you’ve all done all the requisite multiplication is by using FOIL.

Example: Expand and simplify $(x + 3)(x - 4)$.

FOIL is an acronym that stands for First, Outer, Inner, Last. First tells us to multiply the first terms in each bracket together; Outer tells us to multiply the outer two terms; Inner tells us to multiply the inner two terms; Last tells us to multiply the last terms in each bracket.

A good way to track the multiplication you’re doing is by drawing a line between each term once you’ve multiplied it. Here, we’ve added a red line to account for each multiplication.

Once you have 4 red lines, you have completed the requisite multiplications. Now we need to add together the results of those multiplications:

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

Finally, we must collect like terms to satisfy the “simplify” part of the question. We get the final answer to be

$x^2 - x - 12$.

Naturally, as you get better you won’t have working out as extensive as this for each expansion, but it is definitely a good idea to keep track of the expansion by adding in the (red) lines as you go along.

Example: Expand and simplify $(m + 8)(6 - m)$.

## Example Questions

#### 1) Expand $9pq\left(2 - pq^2 - 7p^4\right)$.

We need to multiply everything inside the bracket by $9pq$. We get

$9pq(2 - pq^2 - 7p^4)=9pq \times 2 - 9pq \times pq^2 - 9pq \times 7p^4$

$=18pq -9p^{2}q^3 - 63p^{5}q$.

So, the result of the expansion is $18pq -9p^{2}q^3 - 63p^{5}q$.

#### 2) Expand and simplify $(y-3)(y-10)$.

We need to make sure that we multiply everything in the left bracket by everything in the right bracket. By using FOIL or some other method of remembering to do every multiplication, we get

\begin{aligned}(y-3)(y-10)&=y\times y+y\times(-10)+(-3)\times y +(-3)\times(-10) \\ &=y^2 -10y -3y +30\end{aligned}

Then, collecting like terms we get the result of the expansion to be

$y^2 -13y + 30$.

#### 3) Expand and simplify $(m + 2n)(m - n)$.

We need to make sure that we multiply everything in the left bracket by everything in the right bracket. By using FOIL or some other method of remembering to do every multiplication, we get

\begin{aligned}(m+2n)(m-n)&=m\times m+m\times(-n)+2n\times m +2n\times(-n) \\ &=m^2 -nm +2nm - 2n^2\end{aligned}

Then, collecting like terms we get the result of the expansion to be

$m^2 + nm - 2n^2$.

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