Expanding Binomial Products
Every question with a fully worked solution.
- Expanding Binomial Products - Video - expanding brackets - binomials Watch
Theory
To expand a binomial product such as \((x + 4)(x + 9)\), multiply every term in the first bracket by every term in the second, then collect like terms. This Year 9 maths page shows the area model, the FOIL order, negative terms, a squared bracket and a bracket with three terms.
A binomial is an expression with two terms, such as \(x + 4\) or \(5 - 2x\). A binomial product is two binomials multiplied together, such as \((x + 4)(x + 9)\).
To expand it, multiply each term in the first bracket by each term in the second. Two terms times two terms gives four products. Then collect the like terms, which are usually the two middle products.
The FOIL order is one way to get all four: First terms, Outer pair, Inner pair, Last terms. It is the distributive law used twice.
A squared bracket such as \((x + 7)^2\) means \((x + 7)(x + 7)\). It is expanded the same way.
Every term times every term:
The same thing as the distributive law used twice:
How to expand two brackets
- List the terms of each bracket with their signs.
- Multiply every term in the first bracket by every term in the second: First, Outer, Inner, Last.
- Write out all the products. A squared bracket is written out twice first.
- Collect like terms and write the highest power first.
- Check by substituting a small number such as \(x = 1\).
Draw a rectangle \(x + 9\) wide and \(x + 4\) tall. Each piece is one product:
| \((x + 4)(x + 9)\) | \(=\) | \(x^2 + 9x + 4x + 36\) |
| \(\) | \(=\) | \(\boldsymbol{x^2 + 13x + 36}\) |
The two middle pieces are like terms, so they collect.
(i) \((m - 7)(m + 3)\) and
(ii) \((y - 8)^2\).
(i) The terms are \(m\) and \(-7\), and \(m\) and \(3\). The minus travels with the \(7\):
| \((m - 7)(m + 3)\) | \(=\) | \(m^2 + 3m - 7m - 21\) |
| \(\) | \(=\) | \(\boldsymbol{m^2 - 4m - 21}\) |
(ii) A square means the bracket times itself. Write it out twice:
| \((y - 8)^2\) | \(=\) | \((y - 8)(y - 8)\) |
| \(\) | \(=\) | \(y^2 - 8y - 8y + 64\) |
| \(\) | \(=\) | \(\boldsymbol{y^2 - 16y + 64}\) |
\((-8) \times (-8) = +64\). The answer is not \(y^2 + 64\) or \(y^2 - 64\).
The terms are \(5\) and \(-2x\), and \(3x\) and \(4\). Multiply numbers and pronumerals each time:
| \(5 \times 3x\) | \(=\) | \(15x\) |
| \(5 \times 4\) | \(=\) | \(20\) |
| \(-2x \times 3x\) | \(=\) | \(-6x^2\) |
| \(-2x \times 4\) | \(=\) | \(-8x\) |
Collect the \(x\) terms and write the highest power first:
| \(\text{product}\) | \(=\) | \(-6x^2 + 15x - 8x + 20\) |
| \(\) | \(=\) | \(\boldsymbol{-6x^2 + 7x + 20}\) |
The order of the brackets does not matter: \((3x + 4)(5 - 2x)\) gives the same answer.
Same rule, more terms: \(2 \times 3 = 6\) products. First the \(x\), then the \(-3\):
| \(x(x^2 + 2x + 5)\) | \(=\) | \(x^3 + 2x^2 + 5x\) |
| \(-3(x^2 + 2x + 5)\) | \(=\) | \(-3x^2 - 6x - 15\) |
Add the two rows and collect like terms:
| \(\text{product}\) | \(=\) | \(x^3 + 2x^2 - 3x^2 + 5x - 6x - 15\) |
| \(\) | \(=\) | \(\boldsymbol{x^3 - x^2 - x - 15}\) |
Check with \(x = 1\): \((-2)(8) = -16\), and \(1 - 1 - 1 - 15 = -16\).
Common pitfalls
Frequently asked questions
How do you expand two brackets?
Multiply each term in the first bracket by each term in the second bracket, then collect the like terms. Two terms times two terms gives four products.
What does FOIL stand for?
First, Outer, Inner, Last. It is an order for finding the four products when expanding two brackets.
How do you expand a squared bracket?
Write the bracket out twice and expand as normal. For example \((x + 7)^2 = (x + 7)(x + 7) = x^2 + 14x + 49\).
How does an area model help with expanding?
Draw a rectangle with the two brackets as its side lengths. Split it into pieces; the area of each piece is one of the products, and the total area is the expansion.