>> Welcome to the Cypress College Math Review on Multiplying Polynomials.
Objective 1: Types of Polynomials.
A monomial is a polynomial with one term.
Some examples of a monomial would be 7x squared, negative 3ab, and negative x cubed y squared.
A binomial is a polynomial with two terms.
Some examples of a binomial would be 6x minus 7, and 3a squared plus 2b.
And finally a trinomial is a polynomial with three terms.
And an example of a trinomial would be negative 3x squared y plus 5xy minus 2.
Objective 2: Multiply Two Monomials.
To multiply two monomials, the first step will be to multiply their coefficients together.
Then the second step will be to use the product rule of exponents
to find the exponent of the corresponding variables.
So recall that the Product Rule of Exponents tells us that for multiplying two things
that have the same base, we are able to add their exponents together.
So let's go ahead and multiply these two monomials together.
3x squared times 5x cubed.
So the first thing that we should do is multiply their coefficients together,
so then we would multiply together 3 times 5 and then we'll be multiplying the variables together
so then we will be multiplying together x squared times x cubed.
OK, so multiplying 3 times 5 that will give us 15.
And then now looking at the variables notice that we're multiplying two things
that have the same base of x, so then according to the product rule
of exponents we're able to add their exponents together.
So then we get a final answer of 15x to the 5th.
For our next example we'll multiply the two monomials --
negative 2x squared y by 7x cubed y squared.
So again, the first thing that we'll do is multiply the coefficients together.
Negative 2 times 7 and then notice here we have both x's and y's.
So first we'll multiply the x's together and then we'll also multiply the y's together.
So from here multiplying negative 2 times 7, that will give us negative 14.
And then again using the product rule of exponents we'll be able
to add the exponents together for the x's.
And then looking at the y's, this first y looks like it doesn't have an exponent
but there's actually an invisible exponent of 1 there.
So again similarly we'll be able to add the exponents of the y's together as well.
So then from here we get a final answer of negative 14x to the 5th, y to the 3rd.
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Objective 3: Multiply Any Polynomial by a Monomial.
To multiply a polynomial by a monomial, you will use the distributive property
to multiply the monomial by each term of the polynomial.
So for this example we want to multiply the monomial 3x to the trinomial 2x
to the 4th minus 6x cubed plus 3x squared.
So to multiply these two together, we will distribute the 3x to every term in the trinomial
and when we distribute what we will be doing is multiplying
that monomial 3x to every term in the trinomial.
So let's go ahead and distribute.
So distributing we will get 3x times 2x to the 4th
and then we have a minus 3x times 6x to the third.
And then finally plus 3x times 3x squared.
Next step is to just multiply these monomials together.
So looking at the first two monomials, multiplying those two.
3 times 2 gives us 6x times x to the 4th, again adding the exponents will give us x to the 5th.
Looking at the next two monomials --
negative 3 times 6 that will give us a minus 18, x times x to the 3rd.
Again, adding the exponents will give us x to the 4th.
And then finally looking at the last two monomials, 3 times 3 will give us plus 9.
And x times x squared adding the exponents will give us x cubed.
For this example we want to multiply the monomial negative 2a squared b times the
binomial 4a cubed b squared minus 3a b squared.
So to multiply we will distribute the monomial to both terms in the binomial.
Again, when we distribute what we're really doing is multiplying the monomial
to every term in the binomial.
So distributing will get negative 2a squared b times 4a cubed b squared.
Then we have a minus, distributing again we'll have negative 2a squared b times 3ab squared.
And all there's left to do now is to multiply.
So multiplying the negative 2a squared b times the 4a cubed b squared,
negative 2 times 4 gives us negative 8.
a squared times a cubed, adding the exponents of the a's will give us a to the 5th.
And then b times b squared adding the exponents of the b's will give us b cubed.
Notice here that we have two negatives, so then this will be come a plus.
So then 2 times 3 will give us 6, a squared times a adding the exponents
of the a's will give us a cubed and then b times b squared,
adding the exponents of the b's will give us b cubed.
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Objective 4: Multiply Two Binomials.
To multiply two binomials we will use the distributive property twice or a common acronym
that we use to multiply two binomials together is the acronym FOIL.
Where F will tell us to multiply the First terms together.
O will tell us to multiply the Outer terms together.
I will tell us to multiply the Inner terms together.
And then L tells us to multiply the Last terms together.
So here if you want to multiply the two binomials x plus 7 times 2x minus 3,
we will follow FOIL.
So F tells us to multiply the first terms together,
so then we'll multiply together x times 2x to give us 2x squared.
Then we should multiply the outer terms together that would be x times negative 3
to give us minus 3x and then we want to multiply the inner terms together 7 times 2x will give us
a positive 14x and then we want to multiply together the last terms,
positive 7 times a negative 3 will give us negative 21.
And then from here notice that we have some like terms that we can combine --
the two middle terms -- the negative 3x and the 14x,
we can combine by adding their coefficients together.
So then from here we'll get 2x squared then combining the negative 3x
with the 14x will give us plus 11x and then minus 21.
For this example we'll multiply together the two binomials 3x minus 5y times 2x plus 7y.
So since we're multiplying two binomials together, we will follow FOIL.
So F will tell us to multiply the first terms together.
So then we'll multiply together 3x times 2x to give us 6x squared.
And then we should multiply the outer terms together 3x times 7y.
That will give us plus 21xy.
And then we should multiply the inner terms together.
The negative 5y times a positive 2x will give us negative 10xy
and then we should multiply the last terms together the negative 5y times the positive 7y
to give us negative 35y squared.
And then notice we have two like terms that we can combine.
The 21xy and the negative 10xy are like terms.
So we can combine those two by adding their coefficients together.
So from here we'll get 6x squared combining the two like terms we'll get plus 11xy
and then we have minus 35y squared.
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Objective 5: Multiply the Sum and Difference of the Same Two Terms.
For here, let's say we want to multiply together a plus b times a minus b. Notice
that these two binomials are very similar,
except one has a plus and the other one has a minus.
So let's see what will happen here.
So since we're multiplying two binomials we will follow FOIL.
So we should multiply the first terms together to get a squared,
multiplying the outer terms together we'll get minus ab,
multiplying the inner terms together we'll get plus ab
and then multiplying the last terms together we'll get minus b squared.
So looking at what we have, notice again those two middle terms are like terms
and we can combine those two together.
But when we combine those two together they're just going to eliminate each other,
so then we're just left with a squared minus b squared.
So when you're multiplying two binomials of this form --
a plus b, times a minus b that's just going to multiply to become a squared minus b squared.
For this example we want to multiply together x plus 4 times x minus 4.
So notice that this is of the form a plus b times a minus b. So then we'll know
that this will multiply to become a squared minus b squared.
So then we just need to identify what is our a?
And what is our b?
So notice here that our a will be equal to x, and our b will be equal to 4.
So x squared just becomes x squared minus 4 squared gives us 16.
Here we want to multiply 2x plus 5y times 2x minus 5y.
So notice that this is of the form a plus b times a minus b. So we know
that this will multiply to be of the form a squared minus b squared.
So again we just have to identify what our a is and what our b is.
So for this example our a will become 2x and our b is 5y.
So 2x, all of that squared becomes 4x squared minus and then squaring 5y,
5 squared is 25, and then we have y squared.
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Objective 6: Squaring a Binomial.
So let's say we have the binomial a plus b, which we wish to square.
So if you want to square the binomial a plus b,
that will just be multiplying a plus b times a plus b.
So since we're multiplying two binomials we will follow FOIL.
So we should multiply the first terms together to get a squared,
then we should multiply the outer terms together to get plus ab,
then we should multiply the inner terms together to get plus ab,
and then multiplying the last terms together will get plus b squared.
Now notice those two terms in the middle are like terms so we are able to combine those.
So then we get a squared plus 2ab plus b squared.
So if you have something of the form a plus b quantity squared,
we know that that will multiply out to be a squared plus 2ab plus b squared.
Now let's do a similar example but now we want to square a minus b. So if you want
to square a minus b that is the same as just multiplying a minus b,
times a minus b, and again we will follow FOIL.
So multiplying the first terms together we get a squared,
multiplying the outer terms together we get minus ab,
multiplying the inner terms together we get minus ab
and then multiplying the last terms together we get plus b squared.
And again we have two terms that are like terms,
so combining those two middle terms we get a squared minus 2ab plus b squared.
So if you have something of the form a minus b quantity squared, that will multiply
out to be a squared minus 2ab plus b squared.
For this example we wish to square the binomial a plus 7.
So notice this is of the form a plus b quantity squared.
So we know that this will multiply out to be a squared, plus 2ab, plus b squared.
So then we just need to figure out what our a is and what our b is.
For this example our a is just going to be a. And our b is 7.
So simplifying here we'll get a squared plus 2 times 7 gives us 14a
and then plus 7 squared gives us 49.
For this example we wish to square the binomial 3x minus 4.
So notice that this is of the form a minus b quantity squared, so this will multiply
out to be a squared minus to 2ab plus b squared.
So to figure out this problem, again all we have to do is identify what is our a
and what is our b. So for this one our a is 3x and our b is 4.
So simplifying, 3x squared, well 3 squared is 9.
And then x squared is just x squared.
And then we have minus 2 times 3 gives us 6 times 4 gives us 24.
So then we have minus 24x plus 4 squared will give us plus 16.
Pause the video and try these problems.
Objective 7: Multiply Any Two Polynomials.
To multiply any two polynomials, the first step will be to multiply each term
of the first polynomial to every term in the second polynomial.
Then step two will be to combine any like terms.
So if you wish to multiply together 2x plus 5 times x squared plus 4x minus 3,
we will first have to distribute the 2x to every term in the second polynomial.
And then we will distribute the 5 to every term in the second polynomial.
So again, the first thing we should do is distribute the 2x
to every term in the second polynomial.
So doing that 2x times x squared will give us 2x cubed, 2x times 4x will give us plus 8x squared.
And then 2x times negative 3 will give us negative 6x.
So now that we're done distributing the 2x, the next step is to distribute the 5
to every term in the second polynomial.
So then 5 times x squared will give us plus 5x squared.
5 times 4x will give us plus 20x.
And then 5 times a negative 3 will give us minus 15.
So now that we're done distributing, now we're going to combine any like terms.
So there are no like terms with the 2x cubed.
So I'll just bring him down.
But here with the 8x squared that's like terms with the 5x squared.
So I can combine those two to get plus 13x squared
and then the negative 6x is like terms with the 20x.
So then I can combine those two to get plus 14x.
And then lastly we have that minus 15.
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Objective 8: Multiply More than Two Polynomials.
To multiply more than two polynomials all we'll do is just multiply two polynomials at a time.
So notice for this example we want to multiply together three polynomials.
But again all we're going to do is just work with two polynomials at a time.
So the first two that we're going to be working
with will be multiplying together x plus 1, times x plus 2.
So notice that's just multiplying together two binomials, so we're going to FOIL.
So first thing we do is multiply the first terms together to get x squared.
And then we multiply the outer terms together to get plus 2x.
Multiplying the inner terms together we get plus x
and then multiplying the last terms together we get plus 2.
Hey but then remember we still have to multiply that to the 2x minus 5.
So before we multiply, looking at that first polynomial we can simplify it a little bit more
since we have some like terms there we can combine the 2x with the x,
so doing that first we'll get x squared plus 3x plus 2.
And then that we'll multiply to the 2x minus 5.
So now if we wish to multiply these two polynomials together,
what we'll have to do is distribute each term of the first polynomial
to every term in the second polynomial.
So the first thing that we'll do is multiply the x squared
to the each term, in that second polynomial.
So x squared times 2x gives us 2x cubed,
then x squared times negative 5 will give us minus 5x squared.
So then the next step will be to distribute the 3x to every term in the second polynomial.
So then 3x times 2x will give us plus 6x squared
and then 3x times a negative 5 will give us minus 15x.
And then the last thing we have to distribute is the 2.
So then we'll distribute the 2 to every term in the second polynomial, so 2 times 2x gives us 4x
and then 2 times negative 5 will give us negative 10.
So now that we're done distributing, all we have to do is combine like terms.
So there are no like terms to combine with the 2x cubed, so we'll just bring that down.
But here the negative 5x squared is like terms with the 6x squared.
So combining those two we'll get plus x squared
and then the negative 15x is like terms with the 4x.
So then that will combine to give us negative 11x, and then we have the minus 10.
Pause the video and try these problems.