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Multiplying Polynomials

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Today we're going to talk about multiplying polynomials. So we'll do some examples multiplying different kinds of polynomials and show how you multiply using FOIL or other types of distribution. So for my first example, I'm going to multiply the monomial negative 4y by the binomial 2y to the 3rd plus 5xy to the 4th. So I'll do this by distributing the negative 4y to both terms on the inside of the parentheses.

So I'll start by multiplying negative 4 times 2, which is going to give me a negative 8. And y times y to the 3rd-- since the variables are the same, I can multiply them and add the exponents. So this will become y to the 4th.

Then I've got negative 4y times 5xy to the 4th. So I'll start again by multiplying my coefficients. Negative 4 times a positive 5 will give me a negative 20. y times xy to the 4th-- I can't combine the y variable and the x variable, so I'll simply bring down the x variable. And then I can combine the y times y to the 4th, which will give me y to the 5th.

Now, my answer in standard form should have the term with the biggest degree first. The degree of this first term is 4. But the degree of the second term is going to be 5 plus 1, which is 6. So I need to write my answer with this term first. So this will become negative 20xy to the 5th. And then I'll have my negative 8y to the 4th. So this is my final answer.

So for my second example, I've got the monomial 2x squared multiplied by the polynomial 3x to the 7th minus 5x to the 4th minus 3x plus 1. So I'm going to start again by distributing. I'm going to multiply 2x squared times 3x to the 7th. My coefficients multiplied will give me 6. x squared times x to the 7th is going to give me x to the 9th. I can add the exponents because the variables are the same.

Then I have 2x squared times negative 5x to the 4th. I'll multiply my coefficients, which will give me a negative 10. And multiplying by variables will give me x to the 6th. Then I have 2x squared times negative 3x. Multiplying my coefficients will give me negative 6. x squared times x will give me x to the 3rd. And finally, 2x squared multiplied by 1 is going to simply give me positive 2x squared.

Now, this is my final answer, because my terms are already in order of degree in decreasing order. So I know that they're written in standard form. So again, this is my final answer.

For my third example, I'm multiplying two binomials together. And so I know that to multiply two binomials I should use the FOIL process, which means that I'll multiply my first two terms together, followed by my outside terms, then my inside terms and my last terms. I can also thinkable FOIL as first distributing 3x squared to both terms in this parentheses and then distributing the 2 to both terms in that parentheses. Either way, I'll be able to find my answer.

So let's start by multiplying 3x squared times 5x. Multiplying the coefficients will give me 15. x squared times x will give me x to the 3rd. 3x squared times negative 7-- multiply my coefficients will just give me negative 21. And I'll bring down my x squared.

Then I have 2 times 5x. That's going to give me 10. And bring down the x. And finally, 2 times negative 7 will give me a negative 14. So I check to make sure that this is in standard form, meaning that my terms are in order of degree decreasing. And they are. So this is my final answer.

So for my last example, I've got x minus 2 to the 3rd power. I know that something to the 3rd power means multiplying it by itself 3 times. So I'm going to start by writing this as x minus 2 times x minus 2 times x minus 2.

So now I'm going to start by multiplying my first two binomials together. So that is going to give me x squared minus 2x minus another 2x plus 4. And this can be simplified by combining my two middle terms. And that will give me x squared minus 4x plus 4.

And now I'm going to multiply this trinomial by my last binomial. I'm going to do that by multiplying each of the terms in this trinomial by both terms in the binomial. So I've got x squared times x, which will give me x to the 3rd, x squared times minus negative 2, which would give me negative 2x squared.

Then I've got negative 4x times x, which is going to give me negative 4x squared. And I'm going to write it underneath my other x squared to make it easier to simplify. And I've got negative 4x times negative 2, which will give me a positive 8x.

Finally, I've got 4 times x, which is going to give me a positive 4x, and 4 times negative 2, which will give me a negative 8. So I've got my six terms. And I'm going to combine together any like terms.

X to the 3rd cannot be combined. However, negative 2x squared and the negative 4x squared will give me negative 6x squared. 8x and 4x will give me 12x. And finally, I'll have my minus 8. I'm checking to see that this is written in standard form. And it is. And so this is my final answer.

Let's go over our key points from today. When multiplying polynomials, you need to multiply each term in the first polynomial by each term in the second polynomial. You should combine any like terms to simplify the polynomial. And finally, you should write the polynomial in standard form, ordering the terms by the degree from highest to lowest.

So I hope that these key points and examples helped you understand a little bit more about multiplying polynomials. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.