WEBVTT
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In this video, we'll discuss multiplying polynomials. When multiplying a monomial
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times a polynomial that is not a monomial, we use the distributive property. Let's look
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at these two examples. We're going to begin by multiplying this monomial times this trinomial
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using the distributive property. We multiply negative six x times each of these three terms. This will give us negative six x
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times x squared plus a negative six x times negative three x plus a negative six x times negative six. So we've
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distributed the negative six x to each of these three terms. Now let's simplify the expression.
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Negative six x times x squared. x times x squared using the product rule for exponents gives us x to
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the one plus two or x to the third power. So this term is negative six x to the third power. Here
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we have negative six times negative three which is a positive eighteen, and x times x
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is x to the one plus one, or two, x to the second power. So this will be plus eighteen
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x to the second power or x squared. And in this third term, we have negative six times negative
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six which is a positive thirty-six, and we have x being multiplied times thirty-six, so thirty-six
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x. This leaves us with the trinomial negative six x cubed plus eighteen x squared plus thirty-six x.
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In this example, we have a monomial a b multiplied times this binomial. Again,
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we'll use the distributive property. We're taking a b, multiplying it times the first term which
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is two a squared b, and then plus a b multiplied times the second term three a b squared.
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Now let's simplify these expressions. a b times two a squared b.
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a times a squared is going to be a to the one plus two, or a to the third power. And b times
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b is going to be b to the one plus one, or b to the second power. This gives us two a to the third
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b to the second. And here in our second term, we have a times a giving us a to the second power,
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b times b squared giving us b to the third power. So this term will be three a squared b to the third power.
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The distributive property can also be used to multiply binomials. Let's look at this example.
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x plus one times x plus three. We have a binomial times a binomial. We'll begin
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by distributing this binomial to each of the two terms in the second binomial. That will give us x
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times x plus one, and then distributing to the second term gives us plus three times x plus one.
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Now simplifying this expression, we'll distribute again. We have a monomial times a binomial. This gives us x times x
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which is x squared plus x times one which is x, and distributing three to x plus one gives us three x
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plus three. Finally, combining like terms, we see that we have x to the first power here and an x to the first power here.
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So we can combine these two terms, one x plus three x to give us four x, and the other two terms
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remain the same. So we have x squared plus four x plus three as the product of these two binomials.
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Let's continue with a couple of more examples. We're going to multiply x minus six times x plus eight. So we'll distribute
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this first binomial to each term in the second binomial. This gives us x times x minus six
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plus eight times x minus six. And now again using the distributive property,
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this gives us x squared from x times x and minus six x comes from x times a negative six.
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Distributing here, we have eight times x, or eight x, and eight times negative six, giving us negative
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forty-eight. And finally looking for our like terms, they are here. Negative six x plus eight x
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gives us two x. And the other two terms remain the same. They don't have other like terms. So
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x squared plus two x minus forty-eight is the product of these two binomials.
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In our next example, we have three w minus two times w minus five. Again using the distributive
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property, we will distribute three w minus two to each term in the second binomial, giving us w
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times three w minus two minus five times three w minus two. And now distributing the monomial
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into the binomial gives us w times three w, or three w squared. w times a negative two gives us minus two w.
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A negative five times three w gives us minus fifteen w. And a negative five times a negative two gives us plus ten.
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Combining like terms here, negative two w minus fifteen w gives us negative seventeen w.
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So the final answer here is three w squared minus seventeen w plus ten.
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Our next example is four a plus five b times four a minus five b.
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Again, let's use the distributive property. We're going to distribute this binomial
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to each term in the second binomial, giving us four a times four a plus five b
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minus five b times four a plus five b. Using the distributive property again, we have four a
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times four a which is sixteen a squared. And four a times five b gives us twenty
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a b. Distributing negative five b into this binomial gives us negative five b times four a,
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or minus twenty a b. Notice this could be written as b a or a b, but I chose to write it as a
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b to show that it is similar to this term here in variables. And we have minus five b
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times positive five b giving us minus twenty-five b squared. Now let's combine our like terms.
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Positive twenty a b minus twenty a b. When I add positive twenty to negative
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twenty I get zero a b, and zero times anything is just zero. So these terms
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do not appear in our final answer, leaving us with sixteen a squared minus twenty-five b squared.
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In this next example, we have x squared plus one times x squared plus nine.
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Distributing x squared plus one to each term in our second binomial
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gives us x squared times x squared plus one plus nine times x squared plus one.
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Distributing again gives us x squared times x squared which is x to the fourth because two plus two is four.
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x squared times one gives us x squared. Nine times x squared gives us nine x squared.
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And nine times one gives us nine. And let's look for our like terms. We have an x squared
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and a nine x squared. Adding those coefficients together gives us ten x squared. And so this
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leaves us with x to the fourth plus ten x squared plus nine as our simplified answer.
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Now let's look at our last problem. Write the expression two a plus b squared as a polynomial in standard form.
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So we need to think about what this means. When we have an exponential expression, we have a
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factor raised to a power, meaning that factor is repeated in multiplication. So we're taking two a
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plus b, and we are multiplying it times itself two times. So we want to begin by writing
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two a plus b squared as two a plus b times two a plus b. Again, notice that this is a repeated
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factor, two a plus b, and we're multiplying it times itself two times because the power is two.
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And now we'll use the distributive property. Distributing two a plus b to each term in
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the second binomial gives us two a times two a plus b plus b times two a plus b.
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And then distributing here, two a times two a is four a squared. Two a times b is plus two a b.
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b times two a is also two a b. We can rewrite those terms in whichever order
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we like because multiplication is commutative. So we write that as two a b because it's easier to identify this as a like term.
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And we have b times b which is b squared. And now looking at our like terms, two a b plus two a b
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is four a b. And so this gives us four a squared plus four a b plus b squared as our final answer.
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We can use the distributive property to multiply any two polynomials. In this case,
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we're going to multiply a binomial times a trinomial. So we'll use the distributive property
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and distribute the binomial two x minus five to each of the three terms in this trinomial.
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This will give us two x squared times the binomial two x minus five
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plus three x times the binomial two x minus five minus five times the binomial two x minus five.
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Next, we'll use the distributive property to clear our grouping symbols.
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Multiplying two x squared times this binomial gives us two x squared times two x which
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is four x to the third power. And two x squared times minus five gives us minus ten x squared.
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Distributing three x to each of these two terms in this binomial gives us three x times two x
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which is plus six x squared. And three x times a negative five gives us minus fifteen x.
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Distributing the minus five to the binomial to each term gives us negative five times two x,
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or minus ten x. And negative five times negative five gives us plus twenty-five.
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Finally, we're going to combine like terms. My first term is four x to the third power.
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There are no other x to the third powers, so there are no like terms to combine to four x cubed. My
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second term is negative ten x squared, and it can be combined with positive six x squared.
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Negative ten x squared plus six x squared is minus four x squared.
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Negative fifteen x can be combined with negative ten x because they are like terms.
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Negative fifteen x minus ten x is minus twenty-five x. And plus twenty-five does not have any like terms to be added to it.
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This is the final product of this multiplication of these two polynomials.
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In this example, we're going to write the expression t plus three to the third
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power as a polynomial in standard form. Let's recall what this expression means. The factor t
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plus three is being multiplied times itself three times because the exponent is three.
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So that means that t plus three to the third power can be rewritten as t plus three times t plus three
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times t plus three. So we have a binomial being multiplied times itself
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three times. We're going to begin this problem by multiplying these first two binomials to
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each other, simplifying that product, and then multiplying that resulting polynomial
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times the third binomial. So let's go ahead and multiply t plus three times t plus three
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using the distributive property. We get t times t plus three plus three
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times t plus three. And this product of these two binomials will then be multiplied
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times the third binomial t plus three. Let's simplify this expression by using the distributive
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property to clear the grouping symbols. t times t plus three gives us t squared plus three t.
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And then three times t plus three gives us three t plus nine. And this polynomial will be multiplied times the
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binomial t plus three after we simplify. So we can still combine like terms here in this product to be able
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to simplify this polynomial prior to multiplying by t plus three. Three t plus three t is six t. So this gives us t squared
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plus six t plus nine. This is the product of these first two binomials, and then we're multiplying
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this times the third binomial t plus three. Let's use the distributive property again. Distributing
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this trinomial to each of the two terms in the binomial. This will give us t times the trinomial
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which is t squared plus six t plus nine and then plus three times the trinomial
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again which is t squared plus six t plus nine. Now we'll distribute again to clear these grouping symbols.
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Distributing t to each of these three terms in this trinomial gives us t to the third power plus six t squared plus nine t.
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And distributing this three to each term in this trinomial gives us three t squared plus eighteen t
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plus twenty-seven. And finally we need to look for like terms that can be combined.
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The first term is a t to the third power, and there are no other t cubed expressions. So there are no like terms to combine
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with the first term. I have a six t squared as my next term, and I notice that I have
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a plus three t squared here. Six t squared plus three t squared gives me nine t squared.
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My next term is a plus nine t. And I have an eighteen t here. So these like terms
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can be combined to give me twenty-seven t. And my last term is a plus twenty-seven.
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There are no other constants to be added to this. So there are no like terms to be added at this
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point. And our final answer is t cubed plus nine
t squared plus twenty-seven t plus twenty-seven.