Main content

### Course: Algebra (all content) > Unit 10

Lesson 8: Multiplying binomials by polynomials- Multiplying binomials by polynomials
- Multiply binomials by polynomials
- Multiplying binomials by polynomials: area model
- Multiplying binomials by polynomials challenge
- Multiplying binomials by polynomials review
- Multiplying binomials by polynomials (old)
- Multiplying binomials with radicals (old)

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Multiplying binomials with radicals (old)

An old video Where Sal multiplies and simplifies (x²-√6)(x²+√2). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Is there such a thing as a principle cube root?(31 votes)
- , actually, there is.

Although every real number has one and only one real cube root, it has two more roots than are imaginary or complex.

Just a warning: This gets pretty technical. Learn about imaginary numbers first.

Say we have x^3 = 1, solve for x. (This means x is the cube root of 1)

The possible values for x:

x = 1

x = -1/2 + sqrt(3)/2 i

x = -1/2 - sqrt(3)/2 i

This is how the 2nd answer works out, and you

can do the same process to the 3rd answer.

x = -1/2 + sqrt(3)/2 i

x^3 = (-… (readmoreof this comment)(1 vote)

- What's the difference between the principle square root and the normal square root?(5 votes)
- Great question! When someone simply says or refers to a "square root" of a number, they are not specifying whether or not they want to know the positive or negative root of that number. For example:

The square root of 25 = 5 or -5 (because both 5*5 and -5*-5 are equal to 25)

The "principal square root" refers to ONLY the positive or absolute root value of the number. So:

The principle root of 25 = 5

Hope this helps!!(15 votes)

- With distributive property, how do you multiply a monomial by a binomail?(5 votes)
- You simply distribute the monomial through the binomial

Distributive property: a(b+c) = ab+ac

Ex: 3(x+y) = 3x+3y(12 votes)

- How do we know that the negative square root answer is wrong? Why do we only want the principle square root?(6 votes)
- The square root sign (√) means "principal square root". So if that sign is involved in the equation, you only want the principal square root because that's what the equation is asking for. If the equation is quadratic, such as x^2 = 16, then both square roots (here -4 and +4) are valid answers to the numerical equation, although one of them may be nonsensical if you consider what the equation is about. If you are finding a distance, for example, then a negative square root may not provide a sensible answer.(8 votes)

- This question may be in the wrong topic but I cannot find anything on here that talks about ADDING binomials together that include radicals.

an example: √(3-x) + √(x^2-1)

are equations like this able to be added together?(4 votes)- the rules for adding radical expressions say that you can only add them if what's under the radicals is the same.

3√2 + 2√2 = 5√2 , for example.

I also suggest looking at videos about "radicals" on this page:

https://www.khanacademy.org/math/algebra/exponent-equations(6 votes)

- I don't understand FOIL...is it like PEMDAS?(3 votes)
- FOIL is just an acronym that people use when they first start multiplying things like (6+7) (3+5). It stands for First Outer Inner Last. But you do not have to follow that method like PEMDAS. Later on, you will learn the Distributive Property for solving equations like the one above. As you get to more advanced math, you will find yourself using FOIL not as often, but some people like it and it's a good place for teachers to start.(5 votes)

- How do you add two radicals together?(4 votes)
- You can only do it if they are the square root of the same number. So √2 + √5 cannot be simplified, but √3 + √3 can be simplified to 2√3.(3 votes)

- Why is square root 2 x square root 6 = square root 12?(4 votes)
- Ying,

It is easier to explain with square roots that are rational numbers.

The √9 * √4 = √(9*4) = √36 because

√(3*3) * √(2*2) = √(3*3*2*2) = 3*2 = 6

So

√(x) * √(y) = √(xy)

I hope that helps more than it confuses everyone.(3 votes)

- What does FOIL mean?(3 votes)
- F.O.I.L. is a mnenomic that stands for, "First, Outside, Inside, Last." It is a "trick" to help you remember how to multiply two binomials and is really just the distributive method. For an example lets look at (x+4)(x+5).

First + Outside + Inside + Last

First: x * x = x^2

Outside: x * 5 = 5x

Inside: 4 * x = 4x

Last: 4 * 5 = 20

x^2 + 5x + 4x + 20 = x^2 + 9x + 20(4 votes)

- At3:21why is it sqrt2 - sqrt6 and not the other way around? Or would it still be the same either way?(3 votes)
- It's equivalent, sal choose to do it like that so you only have to use one operation symbol i.e. rather than:

-sqrt6 + sqrt2

He choose:

sqrt2 - sqrt6

But both are equivalent(2 votes)

## Video transcript

We're asked to multiply
and to simplify. And we have x squared minus
the principal square root of 6 times x squared plus the
principal square root of 2. And so we really just
have two binomials, two two-term expressions
that we want to multiply, and there's multiple
ways to do this. I'll show you the
more intuitive way, and then I'll show
you the way it's taught in some
algebra classes, which might be a little bit
faster, but requires a little bit of memorization. So I'll show you the
intuitive way first. So if you have
anything-- so let's say I have a times x
plus y-- we know from the distributive
property that this is the same thing as ax plus ay. And so we can do the
same thing over here. If you view a as x squared--
as this whole expression over here-- x squared minus
the principal square root of 6, and you view x plus y
as this thing over here, you can distribute. We can distribute all
of this onto-- let me do it this way-- distribute
this entire term onto this term and onto that term. So let's do that. So we get x squared minus
the principal square root of 6 times this term-- I'll do
it in yellow-- times x squared. And then we have plus
this thing again. We're just distributing it. It's just like they say. It's sometimes
not that intuitive because this is
a big expression, but you can treat it just like
you would treat a variable over here. You're distributing it over
this expression over here. And so then we have x squared
minus the principal square root of 6 times the principal
square root of 2. And now we can do the
distributive property again, but what we'll do is we'll
distribute this x squared onto each of these terms and
distribute the square root of 2 onto each of these terms. It's the exact
same thing as here, it's just you could imagine
writing it like this. x plus y times a is still
going to be ax plus ay. And just to see the pattern, how
this is really the same thing as this up here,
we're just switching the order of the multiplication. You can kind of view it as we're
distributing from the right. And so if you do this, you
get x squared times x squared, which is x to the fourth,
that's that times that, and then minus x squared times the
principal square root of 6. And then over here you
have square root of 2 times x squared, so plus x squared
times the square root of 2. And then you have square root
of 2 times the square root of 6. And we have a negative
sign out here. Now if you take the
square root of 2-- let me do this on the
side-- square root of 2 times the square root of 6, we
know from simplifying radicals that this is the exact same
thing as the square root of 2 times 6, or the principal
square root of 12. So the square root of 2
times square root of 6, we have a negative
sign out here, it becomes minus the
square root of 12. And let's see if we can
simplify this at all. Let's see. You have an x to
the fourth term. And then here you
have-- well depending on how you want to view
it, you could say, look, we have to second degree terms. We have something
times x squared, and we have something
else times x squared. So if you want,
you could simplify these two terms over here. So I have square
root of 2 x squareds and then I'm going to subtract
from that square root of 6 x squareds. So you could view this
as square root of 2 minus the square root of 6, or
the principal square root of 2 minus the principal square
root of 6, x squared. And then, if you want,
square root of 12, you might be able
to simplify that. 12 is the same
thing as 3 times 4. So the square root of 12
is equal to square root of 3 times square root of 4. And the square root of 4, or
the principal square root of 4 I should say, is 2. So the square root of
12 is the same thing as 2 square roots of 3. So instead of writing the
principal square root of 12, we could write minus 2 times
the principal square root of 3. And then out here you have
an x to the fourth plus this. And you see, if you
distributed this out, if you distribute this x
squared, you get this term, negative x squared,
square root of 6, and if you distribute it onto
this, you'd get that term. So you could debate which
of these two is more simple. Now I mentioned
that this way I just did the distributive
property twice. Nothing new, nothing fancy. But in some classes, you will
see something called FOIL. And I think we've done
this in previous videos. FOIL. I'm not a big fan of
it because it's really a way to memorize a process
as opposed to understanding that this is really just from
the common-sense distributive property. But all this is is
a way to make sure that you're multiplying
everything times everything when
you're multiplying two binomials times
each other like this. And FOIL just says, look,
first multiply the first term. So x squared times x
squared is x to the fourth. Then multiply the outside. So then multiply--
I'll do this in green-- then multiply the outside. So the outside terms are x
squared and square root of 2. And so x squared times
square root of 2-- and they are positive-- so
plus square root of 2 times x squared. And then multiply the inside. And you can see
why I don't like it that much is because you
really don't know you're doing. You're just applying
an algorithm. Then you'll multiply the inside. And so negative square
root of 6 times x squared. And then you multiply
the last terms. So negative square root of
6 times square root of 2, that is-- and we
already know that-- that is negative square root of
12, which you can also then simplify to that expression
right over there. So it's fine to use
this, although it's good, even if you do use this, to
know where FOIL comes from. It really just comes from
using the distributive property twice.