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### Course: Algebra 1>Unit 13

Lesson 2: Multiplying binomials

# Warmup: Multiplying binomials

In this article we are going to get some initial practice with multiplying binomials, to prepare you for the Multiply binomials intro exercise.
If you don't know or remember the distributive property clearly enough, we recommend that you check out this lesson.

## Example 1: Expanding $\left(x+2\right)\left(x+3\right)$‍

There are two ways we can think about this operation. Both are equally valid; you can use whichever you feel more comfortable with.

### First method: Area model

We imagine a rectangle whose height is $x+2$ and width is $x+3$, and divide it into four sub-rectangles:
Now we find the area of each sub-rectangle by multiplying its width and height:
Now we know that this is the area of the entire rectangle, which is the expression we are looking for:
${x}^{2}+3x+2x+6$
We can combine the $x$-terms to get a standard trinomial:
${x}^{2}+5x+6$

### Second method: The distributive property

We can apply the distributive property twice to expand the expression:
$\begin{array}{rl}& \phantom{=}\left(x+2\right)\left(x+3\right)\\ \\ & =\left(x+2\right)x+\left(x+2\right)3\\ \\ & =x\cdot x+2\cdot x+x\cdot 3+2\cdot 3\\ \\ & ={x}^{2}+2x+3x+6\\ \\ & ={x}^{2}+5x+6\end{array}$
In any way, we reached the same result! $\left(x+2\right)\left(x+3\right)$ expanded is ${x}^{2}+5x+6$.

Problem 1.1
Expand and combine like terms.
$\left(x+3\right)\left(x+4\right)=$

## Example 2: Expanding $\left(x-4\right)\left(x+7\right)$‍

Why do we have another example? Well, multiplying binomials becomes a little more tricky when subtraction is involved. Let's see how it's done.

### First method: Area model

As always, we draw a rectangle. However, don't forget to put a minus sign on the $4$.
Now we find the area of each sub-rectangle, keeping in mind that the height of the bottom-left rectangle is $-4$, not $4$.
This doesn't make a lot of sense when thinking about actual rectangles and areas, but it works out with the algebra.
Now we add the areas of all the sub-rectangles:
$\begin{array}{rl}& \phantom{=}{x}^{2}+7x+\left(-4x\right)+\left(-28\right)\\ \\ & ={x}^{2}+3x-28\end{array}$

### Second method: The distributive property

We can apply the distributive property twice, making sure to remember that minus sign!
$\begin{array}{rl}& \phantom{=}\left(x-4\right)\left(x+7\right)\\ \\ & =\left(x-4\right)x+\left(x-4\right)7\\ \\ & =x\cdot x+\left(-4\right)\cdot x+x\cdot 7+\left(-4\right)\cdot 7\\ \\ & ={x}^{2}-4x+7x-28\\ \\ & ={x}^{2}+3x-28\end{array}$

Problem 2.1
Expand and combine like terms.
$\left(x-2\right)\left(x+5\right)=$

## Want to join the conversation?

• Wait, does it really matter which order you put it in? For example, for the practice question I instead did x squared plus 15 -8x? or should I put it like x squared -8x plus 15? And if I do get it right, which is the most recommended?
• It does matter because the exponents the x^2 is larger than x and x is larger than the plain number
• Why can't we just use the foil method instead. (First Outer, Inner, and Last
• It is better to know all the methods so that you can expand what you can do. FOIL works great for (x+2)(x-4) but what if later on in math you have to multiply (x^2 + 2x - 3)(x + 5)? FOIL will not get you very far, but box method or distribution method will.
• Isn't this just the FOIL method? First, Outside, Inside, and Last. That mnemonic helped immensely with my understanding of multiplying polynomials.
• This is similar to FOIL, but it is called the double distribution method, you distribute one of the binomials to all the parts of the other binomial. It is important to learn beyond FOIL because it is not limited to just binomials. You could do (x^2+2x+5)(x-4) using the distribution method. x(x^2+2x+5)-4(x^2+2x+5).
• My method for polynonimals is to write the x terms added together. Is there anything wrong with doing this?
• That's how it should be done.

x^2 +/- (Ax + Bx) +/- C.

You have it right, the x terms should be combined.
• Oh wow,took me a few min. to master this like nothing. Very helpful video!
• does it matter what order they are in?