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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\left(x-3\right)$2(x3) are the product of a term (outside the brackets) and a binomial expression (the sum or difference of two terms). So a binomial is a mathematical expression in which two terms are added or subtracted. They are usually surrounded by brackets or parentheses, such as ($x+2$x+2).

Recall that to expand $2\left(x-3\right)$2(x3) 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).


$\left(x+5\right)\left(x+2\right)$(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.



Question 1

Expand and simplify the following:


Question 2

Expand and simplify the following:


Question 3

Expand and simplify the following:


Question 4

Calculate $86\times58$86×58 by first expressing it in the form $\left(80+6\right)\left(60-2\right)$(80+6)(602).


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