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Multiplying binomials by polynomials

Learn how to multiply binomials by polynomials with ease! This lesson breaks down the process into simple steps, using the distributive property to multiply each term. You'll master combining like terms to simplify expressions, turning complex polynomials into manageable math. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

We are multiplying 10a minus 3 by the entire polynomial 5a squared plus 7a minus 1. So to do this, we can just do the distributive property. We can distribute this entire polynomial, this entire trinomial, times each of these terms. We could have 5a squared plus 7a minus 1 times 10a. And then 5a squared plus 7a minus 1 times negative 3. So let's just do that. So if we have-- so let me just write it out. Let me write it this way. 10a times 5a squared plus 7a minus 1. That's that right over here. And then we can have minus 3 times 5a squared plus 7a minus 1. And that is this distribution right over here. And then we can simplify it. 10a times 5a squared-- 10 times 5 is 50. a times a squared is a to the third. 10 times 7 is 70. a times a is a squared. 10a times negative 1 is negative 10a. Then we distribute this negative 3 times all of this. Negative 3 times 5a squared is negative 15a squared. Negative 3 times 7a is negative 21a. Negative 3 times negative 1 is positive 3. And now we can try to merge like terms. This is the only a to the third term here. So this is 50a to the third. I'll just rewrite it. Now we have two a squared terms. We have 70a squared minus 15, or negative 15a squared. So we can add these two terms. 70 of something minus 15 of that something is going to be 55 of that something. So plus 55a squared. And then we also have two a terms. We have this negative 10a, and then we have this negative 21a. So if we go negative 10 minus 21, that is negative 31. That is negative 31a. And then finally, we only have one constant term over here. We have this positive 3. So plus 3. And we are done.