In Exercises 23–34, find each product using either a horizontal o...

Question

In Exercises 23–34, find each product using either a horizontal or a vertical format.

(x−3)(x²+2x+5)

Accepted Answer

Hey, everyone in this problem, we're asked to multiply the following polynomials and simplify the resulting product. We're given the polynomials P -4, multiplied by the second polynomial P squared plus eight, P plus nine. We have four answer choices. Option A P cubed minus four P squared minus 23 P plus 36. Option B P cubed plus four P squared minus 23 P minus 36. Option choice C P cubed minus four P squared minus 31 P plus 36. And option choice D P cubed plus four P squared minus 31 P minus 36. We're gonna start by rewriting the polynomials. We're given, We have P -4. The first polynomial multiplied by the second polynomial P squared plus eight P plus nine. Well, we're multiplying polynomial together. What we wanna do is we wanna take every term of the first polynomial and multiply it to each term of the second polynomial. What do I need? Well, we're gonna take the first term of the first polynomial P and we're gonna multiply it to the first term of the second polynomial P squared. Then we're gonna multiply it to the second term of our second polynomial, eight p. And then we're gonna multiply it to the third term of our second polynomial nine. So we've multiplied the first term to every term of the second polynomial. And we switch to the second term of the first polynomial. So we have negative four, we're gonna multiply it to P squared. We're gonna multiply it to eight p And then we're gonna multiply it to nine. And this is gonna give us six terms in our expanded multiplied expression. So we go ahead and do this. We have the first term of our first polynomial P multiplied by the first term of the second polynomial P squared. Mean that's a first term of our expression. Then we're going to add again the first term of the first polynomial P Multiplied by the second term of the second polynomial eight p. We add first term, first polynomial p multiplied by the last term of the second polynomial nine. Now we're done with that first term of the first polynomial. We switch to the second term of the first polynomial. We're going to add negative four, multiplied by P squared. OK. And make sure you include this negative. It's important to include the sign when you're doing this so that you get the correct result. Then we add that second term of the first polynomial negative four multiplied by the second term of the second polynomial eight p. And we add, and for our final term, we have the second term of the first polynomial negative four, multiplied by the last term of the second polynomial nine. So we have our expression here with our six terms. We've done the multiplication. And now we're left to simplify, simplifying, we have P multiplied by P squared. When we multiply two things with the same base recall that we add the exponents. So we end up with P to the exponent one plus two. OK. same thing for the next term we have p multiplied by eight p. So we get the coefficient of a and then we have P to the exponent one plus one. The next term we have p multiplied by nine. So we're gonna write this as nine p and we have negative four P squared, we have negative four multiplied by eight P is gonna give us negative 32 P And we have negative four multiplied by nine which gives us -36. Let's go ahead and simplify arcs um exponents here and see where that gets us. We have P Q Plus eight p squared plus nine P -4 p squared -32 p -36. Now we're really close, but we do have some like terms. So we have a P squared term with the coefficient of eight, we have a P squared term with the coefficient of negative four. So we want to combine those, we have P cubed, we have eight P squared minus four P squared that gives us plus four P squared, We have plus nine p -32 p Which gives us -23 p. And then we have our negative 36. OK. So now we have a P Q term, a P square term, a P term and a constant term. And everything is simplified as much as it can be. And so this is our resulting expression. If we compare this to the answer choice, we found that after multiplying our polynomials and simplifying the result that the expression is P cubed plus four P squared minus 23 P minus 36 which corresponds with answer choice B thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 23–34, find each product using either a horizontal or a vertical format. (x−3)(x²+2x+5)

Key Concepts

Polynomial Multiplication

Horizontal and Vertical Formats

Combining Like Terms

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