Expanding Triple Brackets

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Expanding Triple Brackets Revision

Expanding Triple Brackets

Expanding triple brackets is just an extension of expanding single and double 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.

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Expanding Triple Brackets

When expanding triple brackets, we just multiply the first two brackets together, then multiply the result with the final bracket.

Example: Expand the following, $\textcolor{red}{(x+1)}\textcolor{limegreen}{(x+2)}\textcolor{blue}{(3x+3)}$

Step 1 – Multiply out the first two brackets (use FOIL)

Expanding Triple Brackets Step By Step Example Multiply Out First Two Brackets

Step 2 – Multiply the result $x^2 +3x+2$ by $(3x+3)$

This requires $\bf{6}$ separate multiplications, shown below.

Expanding Triple Brackets Step By Step Example Multiply First Two Brackets By Final Bracket

Finally we must collect the like terms

$\textcolor{red}{3x^3+3x^2}\textcolor{blue}{+9x^2+9x}\textcolor{limegreen}{+6x+6} = 3x^3 + 12x^2 +15x + 6$

Example 1: Expanding Triple Brackets

Expand and simplify the following, $\textcolor{red}{(x + 1)}\textcolor{limegreen}{(x+3)}\textcolor{blue}{(x - 2)}$ .

[4 marks]

Step 1 – Multiply out the first two brackets (use FOIL)

$\begin{aligned} \textcolor{red}{(x+1)}\textcolor{limegreen}{(x+3)} &=\textcolor{red}{x^2}\textcolor{limegreen}{+3x}\textcolor{purple}{+x}\textcolor{blue}{+3} \\ &=\textcolor{red}{x^2}\textcolor{maroon}{+4x}\textcolor{blue}{+3}\end{aligned}$

Step 2 – Next we must multiply the result $(\textcolor{red}{x^2}\textcolor{maroon}{+4x}\textcolor{blue}{+3})$ by $\textcolor{blue}{(x - 2)}$

$(\textcolor{red}{x^2}\textcolor{maroon}{+4x}\textcolor{blue}{+3})\textcolor{blue}{(x - 2)} = x^3 + 4x^2 +3x -2x^2 -8x-6 \\ = x^3+2x^2-5x-6$

Example 2: Expanding Triple Brackets

Expand and simplify the following, $\textcolor{red}{(a+5)}\textcolor{limegreen}{(2a-1)}\textcolor{blue}{(3+a)}$ .

[4 marks]

Step one – Multiply out the first two brackets (use FOIL)

$\begin{aligned}\textcolor{red}{(a+5)}\textcolor{limegreen}{(2a-1)} & =\textcolor{red}{2a^2}\textcolor{limegreen}{-a}\textcolor{purple}{+10a}\textcolor{blue}{-5} \\ &=\textcolor{red}{2a^2}\textcolor{maroon}{+9a}\textcolor{blue}{-5}\end{aligned}$

Step 2 – Multiply the result $(\textcolor{red}{2a^2}\textcolor{maroon}{+9a}\textcolor{blue}{-5})$ by $\textcolor{blue}{(3+a)}$

$(\textcolor{red}{2a^2}\textcolor{maroon}{+9a}\textcolor{blue}{-5})\textcolor{blue}{(3 + a)} = 6a^2 + 2a^3 +27a +9a^2 -15 -5a \\ = 2a^3 +15a^2 +22a -15$

Expanding Triple Brackets Example Questions

Question 1: Expand and simplify $(m+8)(m-9)(m+1)$

[4 marks]

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,

$m^3-73m-72$

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Question 2: Expand and simplify $(2k-3)(k+4)^2$

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,

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

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Question 3: Expand and simplify $(x+1)(x+2)(x-4)$

[4 marks]

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,

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

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Question 4: Expand and simplify $(a+b)^3$

[4 marks]

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,

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

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Question 5: Expand and simplify $(x-3)(2x+5)(x+2)$

Firstly, we expand the second 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 second two brackets with their expanded version, we can rewrite the original expression:

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

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 $-3$ in the left-hand bracket by everything in the right-hand bracket. Doing so, we get,

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

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

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

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