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Multiplying Monomials and Binomials

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Today we're gonna talk about multiplying monomials and binomials. Remember, a monomial is just an algebraic expression with one term, and a binomial is an algebraic expression with two terms. So we'll do some examples multiplying monomials and binomials. So here's my first example, I've got 3x squared multiplied by 4x to the fifth plus 2x. So I'm gonna multiply these two expressions by distributing. So I'm gonna start by multiplying 3x squared by 4x to the fifth. I'm gonna multiply all of the factors within each term. So I'll start by multiplying the 3 times the 4 is going to give me 12, and then I'll multiply x squared times x to the fifth which is gonna give me x to the seventh. Next I'm gonna multiply 3x squared by 2x. So again, I'm gonna start by multiplying by 3 times the 2, which is gonna give me positive 6. And then I multiply x squared times x-- or x to the first-- which is gonna give me x to the third. Now, when we are writing a polynomial, the standard form is where the terms of the polynomial are in order by their degree, going from highest to lowest. So the degree of our first term should be higher than the degree of the term behind it. So because seven is the highest degree, and then it goes down to the term with the smallest degree, this expression is in standard form.

Let's do another example. So for my second example I am multiplying 7a times negative 2b squared. So I'm gonna multiply them by multiplying their factors separately. So I'll start by multiplying the coefficients, 7 times negative 2 is gonna give me a negative 14. And then next I'll multiply their variables, so I have a-- or a to the first-- and b squared. Because these are different variables, I can't combine them together by adding their exponents. So I'll simply bring them down, a and then b squared. When we're writing a term with more than one variable, it's general practice to write the variables in alphabetical order. So I'll have my a the first first, followed by my b squared.

From our last example, I've got 5x to the fifth and I'm multiplying that by 3xy minus 8x to the third. So I'm going to start by distributing, multiplying my 5x to the fifth times 3xy. So, I'll start by multiplying by coefficients, 5 times 3 is going to give 15. Then I'm going to multiply x to the fifth times xy. My two variables that are the same, x to the fifth and x-- or x to the first-- I can combine them together by adding the exponents because they're the same variable. So x to the fifth times x to the first is gonna give me x to the sixth. And because my y variable is a different variable, I can't add that exponent to my x, so I'll just bring down the y at the end. Then I'm gonna multiply 5x to the fifth times negative 8x to the third. Again, I'll start by multiplying my coefficients, 5 times negative 8 would give me negative 40. Then I'll multiply my x to the fifth times x to the third, because they have the same variable I can combine them together by adding my exponents, which will give me x to the eighth.

Now I know that the degree of each of these terms is going to be the sum of the degrees of each variable. So this term would have a degree of 6 plus 1 or a degree of 7, and this term would have a degree of 8. And so to write my expression in proper standard form, I need to rearrange the terms so that the term with the highest degree comes first. So I'm going to start by writing my negative 40 x to the eighth term first, followed by my positive 15x to the sixth times y term. And now this is in standard form because my term with the highest degree comes first. In addition, in looking at my variables, they're in alphabetical order, the variable x comes before the variable y. So this is my final answer.

So let's go over our key points from today. When multiplying terms, multiply the coefficients and then multiply the variables. When multiplying variables that are the same, use the product of powers property to add the exponents. And polynomials are written in standard form when the terms are in order by degree from largest to smallest. So I hope that these key points and examples help you understand a little bit more about multiplying monomials and binomials. Keep using your notes, and keep on practicing, and soon you'll be a pro Thanks for watching.