3.07 Binomial expansions

Lesson

What's a binomial?

We've already come across binomial expressions when we looked at how to expand brackets using the  distributive law. Expressions such as $2\left(x-3\right)$2(x3) are the product of a term (outside the brackets) and a binomial expression (the sum or difference of two terms). So a binomial is a mathematical expression in which two terms are added or subtracted. They are usually surrounded by brackets or parentheses, such as $\left(x+7\right)$(x+7).

Recall that to expand $2\left(x-3\right)$2(x3) we use the distributive law: $A\left(B+C\right)=AB+AC$A(B+C)=AB+AC

Now we want to look at how to multiply two binomials together, such as $\left(ax+b\right)\left(cx+d\right)$(ax+b)(cx+d)

Multiplying Two Binomials

When we multiply binomials of the form $\left(ax+b\right)\left(cx+d\right)$(ax+b)(cx+d) we can treat the second binomial $\left(cx+d\right)$(cx+d) as a constant term and apply the distributive property in the form $\left(B+C\right)\left(A\right)=BA+CA$(B+C)(A)=BA+CA. The picture below shows this in action:

As you can see in the picture, we end up with two expressions of the form $A\left(B+C\right)$A(B+C). We can expand these using the distributive property again to arrive at the final answer:

 $ax\left(cx+d\right)+b\left(cx+d\right)$ax(cx+d)+b(cx+d) $=$= $acx^2+adx+bcx+bd$acx2+adx+bcx+bd $=$= $acx^2+\left(ad+bc\right)x+bd$acx2+(ad+bc)x+bd

Let's try an example.

Worked Example

Expand and simplify $\left(x+5\right)\left(x+2\right)$(x+5)(x+2).

Think: We need to multiply both terms inside $\left(x+5\right)$(x+5) by both terms inside $\left(x+2\right)$(x+2).

Do:

 $\left(x+5\right)\left(x+2\right)$(x+5)(x+2) $=$= $x\left(x+2\right)+5\left(x+2\right)$x(x+2)+5(x+2) $=$= $x^2+2x+5x+10$x2+2x+5x+10 Distribute both sets of brackets $=$= $x^2+7x+10$x2+7x+10 Simplify by collecting like terms

Reflect: We chose to split the binomial $\left(x+5\right)$(x+5) up multiply $\left(x+2\right)$(x+2) by both $x$x and $5$5. We could just have easily chosen to split up $\left(x+2\right)$(x+2) and multiply each term by $\left(x+5\right)$(x+5), as follows:

 $\left(x+5\right)\left(x+2\right)$(x+5)(x+2) $=$= $x\left(x+5\right)+2\left(x+5\right)$x(x+5)+2(x+5) $=$= $x^2+5x+2x+10$x2+5x+2x+10 Distribute both sets of brackets $=$= $x^2+7x+10$x2+7x+10 Simplify by collecting like terms

A Visual Interpretation

Let's take the same example as above, $\left(x+5\right)\left(x+2\right)$(x+5)(x+2) and see how this expansion works diagramatically by finding the area of a rectangle.

Notice that the length of the rectangle is $x+5$x+5 and the width is $x+2$x+2. So one expression for the area would be $\left(x+5\right)\left(x+2\right)$(x+5)(x+2).

Another way to express the area would be to split the large rectangle into two smaller rectangles. This way, the area would be $x\left(x+2\right)+5\left(x+2\right)$x(x+2)+5(x+2). Notice that this is the same expression we get after using the distributive property as shown above.

Finally, if we add up the individual parts of this rectangle, we get $x^2+5x+2x+10$x2+5x+2x+10, which simplifies to $x^2+7x+10$x2+7x+10 - the same final answer we found above.

Practice Questions

Question 1

The area of the figure below is $\left(y+3\right)\left(y+6\right)$(y+3)(y+6). We want to find another expression for this area by finding the sum of the areas of the four smaller rectangles.

1. What is the area of rectangle $A$A?

2. What is the area of rectangle $B$B?

3. What is the area of rectangle $C$C?

4. What is the area of rectangle $D$D?

5. Using the areas of each rectangle, write an equivalent expression for the area of the figure.

Question 2

Expand and simplify the following:

$\left(x+6\right)\left(x-12\right)$(x+6)(x12)

Question 3

Expand and simplify the following:

$3\left(y+3\right)\left(y+5\right)$3(y+3)(y+5)

Question 4

Fill in the blanks to make the expression true.

1. $\left(t+6\right)\editable{}=t^2+17t+66$(t+6)=t2+17t+66

Outcomes

9.C1.4

Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.