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10&10a

1.06 Expanding binomial products

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Expansion of binomial products

The distributive law says that for any numbers A,B, and C, A\left(B+C\right)=AB+AC. However A can also be an expression in brackets, and the distributive law still holds.

Consider the expression \left(A+B\right)\left(C+D\right). If we want to expand this using the distributive law we get A\left(C+D\right)+B\left(C+D\right). If we then expand the brackets in both terms we get AC+AD+BC+BD. That is, \left(A+B\right)\left(C+D\right)=AC+AD+BC+BD.

Examples

Example 1

Expand \left(x+10\right)\left(x+5\right)

Worked Solution
Create a strategy

Use the rule \left(A+B\right)\left(C+D\right)=AC+AD+BC+BD.

Apply the idea
\displaystyle \left(x+10\right)\left(x+5\right)\displaystyle =\displaystyle x\times x+x\times 5+10\times x+10\times 5Apply the rule
\displaystyle =\displaystyle x^2+5x+10x+50Simplify the products
\displaystyle =\displaystyle x^2+15x+50Add the like terms
Reflect and check

After using the rule we can then simplify the expression using any of the algebraic rules that we have learned.

Example 2

Expand the following perfect square: \left(x+2\right)^2

Worked Solution
Create a strategy

Use the perfect square expansion: \left(A+B\right)^2=A^2+2AB+B^2

Apply the idea
\displaystyle \left(x+2\right)^2\displaystyle =\displaystyle x^2+2\times 2x+2^2Use perfect squares
\displaystyle =\displaystyle x^2+4x+4Evaluate

Example 3

Expand the following:

\left(m+3\right)\left(m-3\right)

Worked Solution
Create a strategy

Expand using the identity for difference of two squares: \left(A-B\right)\left(A+B\right)=A^2-B^2

Apply the idea

We are given: A=m and B=3.

\displaystyle \left(m+3\right)\left(m-3\right)\displaystyle =\displaystyle m^2-3^2Use the identity
\displaystyle =\displaystyle m^2-9Evaluate the square
Idea summary

We can expand the product of two binomial expressions using the rule

\left(A+B\right)\left(C+D\right)=AC+AD+BC+BD.

There are two special cases of expanding binomials:

  • \left(A+B\right)^2=A^2+2AB+B^2(called a perfect square)

  • \left(A+B\right)\left(A-B\right)=A^2-B^2(called a difference of two squares)

Outcomes

VCMNA332

Expand binomial products and factorise monic quadratic expressions using a variety of strategies.

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