Expanding Brackets Worksheets | Multiplying out Brackets | MME

# Expanding Brackets Worksheet, Questions and Revision

Level 1 Level 2 Level 3

## What you need to know

### Expanding Brackets

Expanding brackets is a key algebra skill that you will require to be able to confidently tackle all sorts of algebra questions. To help you understand expanding brackets you should have a good knowledge of collecting like terms and basic algebra. Understanding the topic expanding brackets will enable you to start to tackle a number of topics including:

As well as the topics above, expanding brackets will enable you to tackle other areas of the curriculum.

### Expanding Single Brackets

The process by which we remove bracket is called expanding (or multiplying out) the brackets. This is the opposite process to factorising.

The first thing to understand about expanding brackets is this: 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.

### Expanding Double Brackets – Foil Method

When expanding double brackets, we need to multiply each of the things in the first bracket is by each of the things in the second bracket. One way of making sure you’ve done all the multiplications required is by using the FOIL method. This works in the following way:

• F – First – This means we multiply together the first term in both brackets.
• O – Outside – This means we multiply the first term by the outer term.
• I – Inner – This means we multiply the second term with the first term (the two closest together on the inside of the double bracket).
• L – Last – This means we multiple the last pair that haven’t been multiplied yet.

You can use the FOIL method when expanding any double bracket and it will always give your answer in the same form, whereby you have to collect the two middle terms together, as these will be like terms in most instances, to arrive at your final answer.

### Example 1: Expanding Single Brackets

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.

### Example 2: Expanding Double Brackets

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

He we can use the FOIL method

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 required multiplications. Now we need to add together the results:

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

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

$x^2 - x - 12$.

### Example 3: Expanding Single Brackets With More Terms

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

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 $15yx^5+25y^{2}x^3+5wy^{3}x^2$.

### Example 4: Expanding Double Brackets

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

### Example Questions

We need to multiply everything inside the bracket by $3xy$, thus

\begin{aligned}3xy(x^2 +2x-8) &= 3xy \times x^2 + 3xy \times 2x + 3xy \times (-8) \\ \\ &= 3x^3y + 6x^2y - 24xy\end{aligned}

We need to multiply everything inside the bracket by $9pq$, thus

\begin{aligned}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\end{aligned}

We need to make sure that we multiply everything in the left bracket by everything in the right bracket.

By using FOIL or another 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

$y^2 -13y + 30$

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

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

First, we can write this as two sets of brackets,

$(2y^2+3x)^2=(2y^2+3x)(2y^2+3x)$

By using FOIL and collecting like terms, we get

\begin{aligned}(2y^2+3x)(2y^2+3x) &=2y^2\times2y^2+2y^2\times3x+3x\times2y^2+3x\times3x \\ \\ &=4y^4+6xy^2+6xy^2+9x^2 \\ \\ &= 4y^4+12xy^2+9x^2\end{aligned}

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