What's a binomial?
We've already come across binomial expressions when we looked at how to expand brackets. Expressions such as $2\left(x-3\right)$2(x−3) are the product of a term (outside the brackets) and a binomial expression (the sum or difference of two terms). So a binomial is a mathematical expression in which two terms are added or subtracted. They are usually surrounded by brackets or parentheses, such as ($x+2$x+2).
Recall that to expand $2\left(x-3\right)$2(x−3) we use the distributive property: $A\left(B+C\right)=AB+AC$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.
Question 1
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 | |
| $=$= | $x^2+7x+10$x2+7x+10 |
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.
Examples
Question 1
Expand and simplify the following:
$\left(x+2\right)\left(x+5\right)$(x+2)(x+5)
Question 2
Expand and simplify the following:
$\left(7w+5\right)\left(5w+2\right)$(7w+5)(5w+2)
Question 3
Expand and simplify the following:
$-\left(x+5\right)\left(x+2\right)$−(x+5)(x+2)
Question 4
Calculate $86\times58$86×58 by first expressing it in the form $\left(80+6\right)\left(60-2\right)$(80+6)(60−2).