[MUSIC PLAYING] Let's look at our objectives for today. We'll first define monomials, binomials, and polynomials. We'll then review the distributive property. We'll do some examples multiplying monomials and binomials. And finally, we'll do some examples multiplying monomials and polynomials.
Let's start by defining monomials, binomials, and polynomials. Here's an example of a monomial-- "mono" meaning "one." A monomial is an exponential expression that consists of one term with non-negative integer exponent. In this term, 4 is the coefficient, x is the base, and 5 is the exponent.
Here is an example of a binomial-- "bi" meaning "two." A binomial is an expression that consists of two monomial terms. And here's an example of a polynomial-- "poly" meaning "many." A polynomial is an expression that consists of one or more monomial terms.
Expressions should always be simplified by combining like terms if possible. In this polynomial, we can combine the like terms 4a and 2a. 4a plus 2a is 6a, so this expression can be simplified to 5a to the third minus 3a squared plus 6a.
Now, let's review the distributive property. The distributive property says that a times b plus c is equal to a times b plus a times c, where the quantity that is outside of the parentheses is multiplied or distributed into every term inside the parentheses. Here's an example. To simplify the expression 7 times x minus 4, we distribute multiplying the 7 to both the x and the negative 4 in the parentheses. 7 times x is 7x, and 7 times negative 4 is negative 28. So we have 7x minus 28.
Let's do an example multiplying a monomial by a binomial. When multiplying a monomial by a binomial, the entire monomial is distributed including the coefficients and the variable powers. We want to multiply 4m to the third by 5m squared plus 2m. Because our variable bases are the same, we can use the product property for exponents to add our exponents together when we multiply the variable powers together.
To multiply, we'll use the distributive property to multiply 4m to the third by 5m squared and 4m to the third by 2m. When we multiply 4m to the third by 5m squared, we can use the commutative property of multiplication to rewrite the product by grouping our coefficients together and our variable powers together. This gives us 4 times 5 times m to the third times m squared. Similarly, when we multiply 4m to the third by 2m, we can again use the commutative property of multiplication to rewrite our product as 4 times 2 times m to the third times m.
Simplifying our first term, we multiply our coefficients. 4 times 5 gives us 20. And then we use the product property of exponents to add our exponents together, which gives us m to the fifth.
Simplifying our second term, we multiply our coefficients. 4 times 2 gives us 8. We then see that our variable m has no written exponent, and we know that that has an implied exponent of 1. So adding our exponents together gives us m to the fourth.
This expression is in standard form because the term with the highest exponent power is written first, the term with the next highest exponent power is written second, and so on. We also call standard form of a polynomial "descending order."
Here is our second example. We want to multiply negative 2x squared times 3x to the sixth minus x. We multiply negative 2x squared by both terms in the parentheses. So first, we have negative 2 times 3, which is negative 6, and x squared times x to the 6, which is x to the eighth-- negative 6x to the eighth.
We then multiply negative 2x squared times negative x. The x here has an implied coefficient of negative 1, so we have negative 2 times negative 1, which is a positive 2, and x squared times x, which is x to the third. So 2x to the third. And our expression is in standard form or descending order, so this is our final answer.
Now, let's do some examples multiplying monomials with polynomials. We have the example below. We need to distribute the 3 on the outside of the parentheses to all three terms in the parentheses. Notice that the 3 does not have any variable bases or exponents, so we only need to multiply the coefficients.
Also, remember that there is an implied negative 1 being multiplied by the b here. So we have 3 times 2a squared, which is 6a squared. Then 3 times a to the third b squared, which is 3a to the third b squared. And finally, 3 times a negative b, which is negative 3b.
We then reorder our terms so that they are in descending order, so we switch the first and the second terms. Our final answer is 3a to the third b squared plus 6a squared minus 3b.
So here is our last example. We want to multiply, as shown in the expression below. We need to multiply the term on the outside, 7x squared y cubed, by each of the three terms inside the parentheses. So we begin by multiplying by the first term in the parentheses. 7 times 2 is 14, and we can also add the exponents of x squared and x to the fifth because their bases are both x. So x squared times x to the fifth is x to the seventh, and y to the third stays unchanged.
We now multiply 7x squared y to the third by our second term in the parentheses. This gives us a coefficient of negative 35. x squared times x squared is x to the fourth, and y cubed times y squared is y to the fifth.
Finally, we multiply by the last term in our parentheses. Our coefficient will be 21, x squared remains unchanged, and y to this third terms y to the third is y to the sixth. And our expression is in standard form or descending order, so this is our final answer.
Let's go over our key points from today. Make sure you get these in your notes so you can refer to them later. A monomial is an exponential expression that consists of one term with non-negative integer exponents. A binomial is an expression that consists of two monomial terms, and a polynomial is an expression that consists of one or more monomial terms. When multiplying a monomial by a polynomial, the entire monomial is distributed-- including coefficients and variable powers.
So I hope that these key points and examples helped you understand a little bit more about multiplying terms using distribution. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
An expression containing two monomial terms.
An exponential expression with non–negative integer exponents.
An expression containing two or more monomial terms.