Expanding Triple Brackets Worksheets | Questions and Revision

# Expanding Triple Brackets Worksheets, Questions and Revision

Level 6 Level 7

## What you need to know

### Expanding Triple Brackets

Previously, we’ve seen how to expand single and double brackets – if you don’t remember, visit (multiplying single double brackets) – so now we’re going to look at expanding triple brackets. Fortunately, this new kind of expansion isn’t really a new kind at all. The ideas are much the same as when we expanded double brackets, it’s just the messier algebra that makes it tricky.

### Take Note:

The following tips will hopefully help you to reduce the amount of errors you make when expanding brackets.

1. When expanding brackets, the number of terms you end up with (before collecting like terms) is the number of terms in 1 bracket multiplied by the number of terms in the other bracket. With triple brackets where we have 2 terms in each, this would be $2\times 2\times 2=6$
2. Having well set out workings that you repeat in the same way for different questions will help you to reduce the number of errors you make in all topics. This becomes more important in topics like expanding triple brackets where you can end up with many algebra terms that can easily be muddled up.

### Example 1: Expanding Triple Brackets

Expand and simplify $(x - 2)(x + 1)(x + 3)$.

To do this, we’re first going to expand the second two brackets, $(x+1)(x+3)$, as this is a familiar process where we use the FOIL method. So, we get

\begin{aligned}(x+1)(x+3)&=x^2+3x+x+3 \\ &=x^2+4x+3\end{aligned}

Now we’ve expanded these two brackets, we can rewrite the original expression, replacing the last two brackets with their expansion. Doing so, we get

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

This is now a double bracket expansion. Sadly, there is no handy acronym like FOIL that we can use in this case. We must remember to multiply every term in the first bracket by every term in the second as shown.

Thus, the simplified result of the expansion is $x^3+2x^2-5x-6$.

### Example 2: Expanding Triple Brackets

Expand and simplify $(a + 5)(2a - 1)(3 + a)$.

Firstly, we expand the second two brackets into a normal quadratic. Using FOIL, we get

\begin{aligned}(2a-1)(3+a)&=6a+2a^2-3-a \\ &=2a^2+5a-3\end{aligned}

Then, replacing the second two brackets with their expanded version, we can rewrite the original expression:

$(a+5)(2a-1)(3+a)=(a+5)(2a^2+5a-3)$

We will now expand these two brackets, doing so, we get

$(a+5)(2a^2+5a-3)=2a^3+5a^2-3a+10a^2+25a-15$

A quick count shows that this expression has 6 terms, which means we’re probably all good so far. What remains is to collect like terms. Doing so, we get the simplified expansion to be

$2a^3+15a^2+22a-15$

### Example Questions

Firstly, we expand the second two brackets into a normal quadratic,

\begin{aligned}(m-9)(m+1)&=m^2+m-9m-9 \\ &=m^2-8m-9\end{aligned}

Then, replacing the second two brackets with their expanded version, we can rewrite the original expression:

$(m+8)(m-9)(m+1)=(m+8)(m^2-8m-9)$

We will now expand these two brackets, first multiplying the $m$ in the left-hand bracket by everything in the right-hand bracket, and then multiplying the 8 in the left-hand bracket by everything in the right-hand bracket. Doing so, we get

$(m+8)(m^2-8m-9)=m^3-8m^2-9m+8m^2-64m-72$

Collecting and simplifying like terms, we get the simplified expansion to be,

$m^3-73m-72$

As $(k+4)^2$ is the same as $(k+4)(k+4)$ the expression can also be written like,

$(2k-3)(k+4)(k+4)$

This then looks like all the other triple bracket expansions we’ve seen, and we can continue as normal. Firstly, we expand the second two brackets into a normal quadratic,

\begin{aligned}(k+4)(k+4)&=k^2+4k+4k+16 \\ &=k^2+8k+16\end{aligned}

Then, replacing the second two brackets with their expanded version, we can rewrite the original expression:

$(2k-3)(k+4)(k+4)=(2k-3)(k^2+8k+16)$

We will now expand these two brackets, first multiplying the 2k in the left-hand bracket by everything in the right-hand bracket, and then multiplying the -3 in the left-hand bracket by everything in the right-hand bracket. Doing so, we get,

$(2k-3)(k^2+8k+16)=2k^3+16k^2+32k-3k^2-24k-48$

Collecting and simplifying like terms, we get the simplified expansion to be,

$2k^3+13k^2+8k-48$

Firstly, we expand the second two brackets into a normal quadratic,

\begin{aligned}(x+2)(x-4)&=x^2+2x-4x-8 \\ &=x^2-2x-8\end{aligned}

Then, replacing the second two brackets with their expanded version, we can rewrite the original expression:

$(x+1)(x+2)(x-4)=(x+1)(x^2-2x-8)$

We will now expand these two brackets, first multiplying the $x$ in the left-hand bracket by everything in the right-hand bracket, and then multiplying the 1 in the left-hand bracket by everything in the right-hand bracket. Doing so, we get

$(x+1)(x^2-2x-8)=x^3-2x^2-8x+x^2-2x-8$

Collecting and simplifying like terms, we get the simplified expansion to be,

$x^3-x^2-10x-8$

We can write the expression given in the question as,

$(a+b)^3=(a+b)(a+b)(a+b)$

This then looks like all the other triple bracket expansions we’ve seen, and we can continue as normal. Firstly, we expand the second two brackets into a normal quadratic,

\begin{aligned}(a+b)(a+b)&=a^2+ab+ba+b^2 \\ &=a^2+2ab+b^2\end{aligned}

Then, replacing the second two brackets with their expanded version, we can rewrite the original expression:

$(a+b)^3=(a+b)(a^2+2ab+b^2)$

We will now expand these two brackets, first multiplying the $a$ in the left-hand bracket by everything in the right-hand bracket, and then multiplying the $b$ in the left-hand bracket by everything in the right-hand bracket. Doing so, we get,

$(a+b)(a^2+2ab+b^2)=a^3+2a^2b+ab^2+a^2b+2ab^2+b^3$

Collecting and simplifying like terms, we get the simplified expansion to be,

$a^3+3ab^2+3a^2b+b^3$

Firstly, we expand the first two brackets into a normal quadratic,

\begin{aligned}(2x+5)(x+2)&=2x^2+4x+5x+10 \\ &=2x^2+9x+10\end{aligned}

Then, replacing the first two brackets with their expanded version, we can rewrite the original expression:

$(2x+5)(x+2)(x-3)=(2x^2+9x+10)(x-3)$

We will now expand these two brackets, first multiplying the $a$ in the left-hand bracket by everything in the right-hand bracket, and then multiplying the $b$ in the left-hand bracket by everything in the right-hand bracket. Doing so, we get,

$(2x^2+9x+10)(x-3)=2x^3 +9x^2+10x-6x^2-27x-30$

Collecting and simplifying like terms, we get the simplified expansion to be,

$2x^3+3x^2-17x-30$

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