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## College Algebra

### Course: College Algebra > Unit 4

Lesson 2: Special products of binomials- Special products of the form (x+a)(x-a)
- Squaring binomials of the form (x+a)²
- Multiply difference of squares
- Multiply perfect squares of binomials
- Special products of the form (ax+b)(ax-b)
- Squaring binomials of the form (ax+b)²
- Binomial special products review

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# Squaring binomials of the form (x+a)²

Sal introduces

**perfect square expressions**. For example, (x+7)² is expanded as x²+14x+49.## Want to join the conversation?

- At2:55, Isn't the binomial (x+a)(x+b) = x^2 + (a+b)x +ab? Thanks.(148 votes)
- Yeah, Sal made a mistake there. It should've been a*b and not b^2(37 votes)

- Sal,

Is 2x equivalent to X^2? Logically, it would seem so: 2-X's or X,X (squared), yet, I am relearning math after decades of "dust." I don't assume anything! lol(7 votes)- 2X = X^2 only if X=2. Then you would have 2(2) = 2(2)

2X means 2 times X or X+X

X^2 means X times X

So, if X = 3, 2X becomes 2(3) = 6 and X^2 become 3(3) = 9

Hope that clarifies things.(19 votes)

- Has Sal released a video on trinomials?(10 votes)
- with an expression like (x+a)^2 why wouldn't you just distribute the exponent to do x^2 + a^2 ?(4 votes)
- Exponents represent repetitive multiplication. Thus, the exponent property that distribute the exponent only works when you have factors (items being multiplied or divided). The expression (x+2)^2 contains terms inside the parentheses. So, the exponent properties do not apply.

To simplify (x+2)^2, you need to use distributive property or FOIL. Or, you can learn the pattern as Sal shows at bout3:00into the video. Squaring a binomial creates a perfect square trinomial.

Hope this helps.(13 votes)

- What if we have this kind of expression
`(3x+2)^2`

, is its form also`x^2+2ax+a^2`

?(4 votes)- Almost; in this case you have a factor of 3 along with x, which you also need to take into account. The general form (without x or numbers) is (a+b)^2 = a^2 + 2ab + b^2. In your example a = 3x and b = 2 (I hope it's not too confusing, the b in the general form is the a in the video).

So then a^2 = (3x)^2 = 9x^2; b^2 = 2^2 = 4; and 2ab = 2*3x*2 = 12x.

Putting it all together:

(3x+2)^2 = 9x^2 + 12x + 4.(11 votes)

- In exponents properties, there's this property for taking a power of a product: (x*y)^n=x^n*y^n. But when taking a power of a sum like in this video, I've noticed that it seems to work differently, and I find that confusing. Could anyone clarify? Why isn't it: (x+y)^2 = x^2+y^2?(9 votes)
- Well using the distributive property, (x+y)^2 gets distributed to (x+y)(x+y) and then either using the FOIL method you proceed ahead or in the common way which leads up to x^2+2xy+y^2. Instead if you do it x^2+y^2 then it simplifies to (x+y)(x-y),as taught in the previous videos by Sal, which is totally different than your question. Hope I was of some help!(0 votes)

- Why doesn't (x+7)^2 just become x^2+7^2 as we learned earlier?(5 votes)
- What you should have learned earlier is that to multiply two binomials you need to use FOIL or extended distribution.

(x+7)^2 = (x+7)(x+7) = x^2+7x+7x+49 = x^2+14x+49

i believe you are trying to apply the power property of exponents: (ab)^n = a^n*b^n. The property only works with factors (items being multiplied / divided). You are trying to apply it to terms (items being added/subtracted).

Hope this helps.(7 votes)

- I don’t find these so called short cuts worth it, you’re likely to misremember it and doing the distributive property takes like 3 more seconds.(4 votes)
- You are right that it doesn't save much time in multiplying the 2 binomials. However, when you get to later lessons on factoring quadratics, it can speed up the process. The pattern is also used in converting quadratic equations to vertex form. You also use it with equations of circles. So, it is worth understanding and learning the pattern.(7 votes)

- I learned a different method, FOIL, does it apply here as well?(5 votes)
- FOIL always works, this is just a shortcut is you specifically have a square.(3 votes)

- What is a binomial?(2 votes)
- A binomial is just two
*terms*that don't combine or cancel out. For instance (3x - 2) is a binomial. However, (-x + 2y + 4) and (x^2 + 2x - 1) are not binomials because they have more than two terms.(12 votes)

## Video transcript

- [Voiceover] Let's see
if we can figure out what x plus seven, let me write
that a little bit neater, x plus seven squared is. And I encourage you to pause the video and work through it on your own. Alright, now let's work
through this together. So we just have to remember, we're squaring the entire binomial. So this thing is going
to be the same thing as: x plus seven times x plus seven. I'm gonna write the second x plus seven in a different color, which is going to be helpful when we actually multiply things out. When we see it like this, then we can multiply these out the way we would multiply any binomials. And I'll first do it
the, I guess you can say, the slower way, but
the more intuitive way, applying the distributive property twice. And then we'll think
about maybe some shortcuts or some patterns we might
be able to recognize, especially when we are squaring binomials. So let's start with just applying the distributive property twice. So let's distribute
this yellow x plus seven over this magenta x plus seven. So we can multiply it by
the x, this magenta x, so it's going to be x, let
me do it in that same color. So it's going to be magenta x times x plus seven plus magenta seven times yellow x plus seven. X plus seven, and now we can apply the distributive property again. We can take this magenta x and distribute it over the x plus seven. So x times x is x-squared. X times seven is seven x. And then we can do it again over here. This seven, let me do
it in a different color, so this seven times that x is going to be plus another seven x and then the seven times the seven is going to be 49. And we're in the home stretch.
We can then simplify it. This is going to be x-squared and then these two middle
terms we can add together. Seven x, let me do this in orange, seven x plus seven x is going to be 14 x plus 14 x plus 49. Plus 49. And we're done. Now the key question is do
we see some patterns here? Do we see some patterns
that we can generalize and that might help us square binomials a little bit faster in the future? Well, when we first looked at
just multiplying binomials, we saw a pattern like x plus a times x plus b is going to be equal to x-squared, let me write it this way, is going to be equal to x-squared plus a plus b x plus b-squared. And so, if both a and
b are the same thing, we can say that x plus a times x plus a is going to be equal to x-squared, and this is the case when we have a coefficient of one on both
of these x's, x-squared's. Now in this case, a and b are both a. So it's going to be a plus a times x, or we can just say plus two a x. Let me be clear what I just did. Instead of writing a plus b, I can just view this as a plus a times x, and then plus a-squared, or that's the same thing as x-squared plus two a x plus a-squared. This is a general way of expressing a squared binomial like this. A squared binomial where the coefficients on both x's are one. We can see that's exactly
what we saw over here. In this, in the example
we did, seven is our a. So we got x-squared right
over there let me circle it. So we have this blue x-squared that corresponds to that over there. And then seven is our a, so two a x , two times seven is 14 x. Notice we have the 14 x right over there. So this 14 x corresponds to two a x, and then finally if a is seven, a-squared is 49. A-squared is 49. So in general if you are
squaring a binomial, you could, a fast way of doing it is
to do this pattern here, and we can do another example real fast, just to make sure that
we've understood things. If I were to tell you what is x minus, I'll throw a negative in here, x minus three squared, I encourage you to pause the video and think about it. Think about expressing
this using this pattern. Well this is going to
be, in this case our a, we have to be careful, our a is going to be negative three, so that is our a right over there. So this is going to be equal to x-squared. Now two a x, let me do it
in the same colors actually, just for fun. So it's going to be x-squared. Now what is two times a times x? A is negative three, so two
times a is negative six. So it's going to be negative six x. So, minus six x, that's two
times a is the coefficient. And then we have our x there. And then plus a-squared. Well if a is negative three, what is negative three
times negative three? It's going to be positive nine. And just like that, when
we looked at this pattern, we were able to very quickly figure out what this binomial squared actually is. And I encourage you, you can do it again, with applying the
distributive property twice to verify that this is
indeed the same thing as x minus three squared.