Expand binomial expressions
What's a binomial?
We've already come across binomial expressions when we looked at how to expand brackets. Expressions such as $$2(x−3) are the product of a term (outside the brackets) and a binomial expression (the sum or difference of two terms). So a binomial is a mathematical expression in which two terms are added or subtracted. They are usually surrounded by brackets or parentheses, such as ($$x+2).
Recall that to expand $$2(x−3) we use the distributive property: $$A(B+C)=AB+AC
Now we want to look at how to multiply two binomials together, such as $$(ax+b)(cx+d).
Multiplying Two Binomials
When we multiply binomials of the form $$(ax+b)(cx+d) we can treat the second binomial $$(cx+d) as a constant term and apply the distributive property in the form $$(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 $$A(B+C). We can expand these using the distributive property again to arrive at the final answer:
| $$ax(cx+d)+b(cx+d) | $$= | $$acx2+adx+bcx+bd |
| $$= | $$acx2+(ad+bc)x+bd |
Let's try an example.
Question 1
Expand and simplify $$(x+5)(x+2) .
Think: We need to multiply both terms inside $$(x+5) by both terms inside $$(x+2).
Do:
| $$(x+5)(x+2) | $$= | $$x(x+2)+5(x+2) |
| $$= | $$x2+2x+5x+10 | |
| $$= | $$x2+7x+10 |
A Visual Interpretation
Let's take the same example as above, $$(x+5)(x+2) and see how this expansion works diagramatically by finding the area of a rectangle.
Notice that the length of the rectangle is $$x+5 and the width is $$x+2. So one expression for the area would be $$(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 $$x(x+2)+5(x+2). Notice that this is the same expression we get after using the distributive property as shown above.
Finally, if we add up the individual parts of this rectangle, we get $$x2+5x+2x+10, which simplifies to $$x2+7x+10 - the same final answer we found above.
Examples
Question 1
Expand and simplify the following:
$$(x+2)(x+5)
Question 2
Expand and simplify the following:
$$(7w+5)(5w+2)
Question 3
Expand and simplify the following:
$$−(x+5)(x+2)
Question 4
Calculate $$86×58 by first expressing it in the form $$(80+6)(60−2).