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1 Algebra
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Australia
Year 10

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

ACMNA230

Factorise algebraic expressions by taking out a common algebraic factor

ACMNA231

Simplify algebraic products and quotients using index laws

ACMNA232

Apply the four operations to simple algebraic fractions with numerical denominators

ACMNA233

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

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