Multiply: (2x−5)(x²−3x−6). (Section 5.2, Example 3)

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Hey, everyone in this problem, we're asked to find the product of the binomial and trinomial. The binomial we're given is 6 X -7 and that's multiplied by the trinomial X squared minus five X minus 11. We're given four answer choices. Option A six X cubed plus 37 X squared plus 31 X plus 77. Option B six X cubed minus 37 X squared minus 31 X plus 77. Option C six X cubed plus 37 X squared minus 31 X plus 77. And option D six X cubed minus 37 X squared plus 31 X plus 77. We're gonna start by rewriting what we were given. So we have our binomial 6 X - and it's multiplied by our trinomial X squared minus five X minus 11. When we're multiplying two expressions together like this, we want to make sure we multiply every term of the first expression by every term in the second expression. So every term in our binomial needs to multiply every term in our trinomial. So starting with six X, we're gonna wanna multiply six X to the first term of the trinomial X square. Then we want to multiply it to the second term of the trinomial negative 5 x and then to the third term of the trinomial negative 11. And then we're done with the first term of our binomial and we move on to the second term of our binomial, which is negative seven and we do the same thing, we multiply it to the X squared to the -5 x and to the -11. So let's go ahead and do that. And I'm gonna write it down here because we're gonna have a lot of terms and I don't want to run out of space. So we take six X, the first term of the binomial multiply it by X squared, the first term of the trinomial. So we have 6 x multiplied by X squared. Then we're gonna add the next room again, the first term of the binomial six X multiplied by the second term of the trinomial negative five X six X multiplied by -5 X. Then we add, we do the same thing. First term of the binomial times the third term of the trinomial 6 x multiplied by -11. Now we've multiplied the first term of the binomial by each of the terms in the trinomial. Now we switch over to the second term of our binomial negative seven multiplied by the first term of the trinomial X squared. Can we add negative seven multiplied by the second term of the tri trinomial negative five X. And then we add -7, multiplied by the last term of the trinomial negative 11. OK. And now we've multiplied each term of the binomial by each term of the trinomial. OK. So we've done the multiplication step now. And what we're left to do is just to simplify starting with the first term six X multiplied by X squared, I get six X. Now when we multiply two things together with the same base, we're called it the exponents act. So six X multiplied by X squared gives us six X to the exponent one plus two. Our next term six X multiplied by negative five X. The coefficient is gonna be negative 30. We have X multiplied by X. So we get X to the exponent one plus one. Our next term 6 x multiplied by -11. This is gonna give us negative 66 X. So we subtract we have negative seven multiplied by X squared. This gives us negative seven X squared, negative seven multiplied by negative five X gives us positive 35 x And -7 multiplied by -11 gives us positive 77. Now we're getting in terms of our simplification, there's a little bit more we can do. So let's go to the next line. We can simplify these exponents. So we have six X to the exponent one plus two that can be rewritten as 6 x cued we have 30 X to the exponent one plus one and it's negative. And so we subtract 30 X squared and all of our other terms already have simplified exponents. So we have minus 66 X minus seven X squared plus 35 X plus 77. Now we simplified our exponents. There's one step left to do. And that is just to simplify these terms. OK? We have a single X cube term which is good, but we have two different terms that contain an X squared. So we want to combine those and similarly with these X terms. So we get six cued, we have negative 30 X squared minus seven X squared. It's gonna use us minus 37 X squared. We have negative 66 X plus 35 X gives us minus 31 X. And finally, our constant term plus 77. And now we have our simplified expression, we have a single X cube term, a single X squared term, a single X term and a single constant term. If we compare this to the answer choices, we found that the product of the binomial and the trinomial is six X cubed minus 37 X squared minus 31 X plus 77 which corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

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Multiply: (2x−5)(x²−3x−6). (Section 5.2, Example 3)

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