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−1)(x²+x+1)

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to multiply the following polynomials and simplify the resulting product. The polynomials were given, we have U -5, multiplied by the second polynomial U squared plus five, U plus 25. We're given four answer choices. Option A U cubed plus five U squared minus 25 U minus 125. Option B U cubed minus five U squared plus 25 U minus 125. Option CU Cubed -125. In option DU cubed plus 125, we're gonna start by rewriting what we've been given. We have the first polynomial U minus five, multiplied by the second polynomial U squared plus five U plus 25. Now, we're asked to multiply these together and when we're multiplying two polynomials together, what we wanna do is take each term of the first polynomial and multiply it to every term of the second polynomial. Now, what do I need? So we're gonna take U the first term of our first polynomial and we're gonna multiply it to the first term of the second polynomial. U squared, then we're gonna take it and we're gonna multiply it to the second term by you. Then we're gonna take it and we're gonna multiply it to the last term. 25. Ok. Well, we're done with that. We're gonna start with the next term. So we're gonna take the second term negative five of our first polynomial and we're gonna multiply it to each of the terms in this second polynomial as well. And that is going to give us six terms. Now, when we do this, what are we gonna get? Well, the first term you multiplied by the first term of the second polynomial U squared, then we're gonna act. OK. The next term is gonna be the first term of the first polynomial U Multiplied by the second term of the second polynomial. five U. So you Multiplied by five U we add and again, first term of the first polynomial multiplied by the wax term of the second polynomial you Multiplied by 25. OK. So we're done with the first term of the first polynomial. We can move to the second term of the first polynomial. We're gonna add, we have negative five and don't miss that minus. OK. It's important to include the sign we wanna have that throughout our entire calculation. So we have negative five multiplied by the first term of the second polynomial U squared plus - multiplied by the next term five U Plus negative five multiplied by the last term 25. OK. So we have our six terms here. Now we've done the multiplication. Now it's just a matter of simplifying this result. Our first term U multiplied by U squared. When we multiply two terms with the same base, we add the exponent. OK. Recall that rule. And so we get you to the exponent 1-plus 2. In the next term, we have five, this constant coefficient U multiplied by U. We get you to the exponent one plus one. OK. Applying that same rule multiplying the same base, add the exponents. In the next term. We have u multiplied by 25. You can write this as 25, we have negative five multiplied by U squared. So minus five U squared, we have negative five multiplied by five U. So negative five multiplied by five gives us a coefficient of negative 25. And so we have negative 25 U for this term In our final term negative five multiplied by 25. This gives us -125. Let's go ahead and simplify arc spits. We have U to the exponent one plus two, gives us U cubed plus five U to the exponent one plus +15 U squared. And then we have the remainder of our terms plus 25 U minus five U squared minus 25 U minus 125. If we look at these terms, we have A plus five U squared minus five U squared, those two are going to add to zero, We also have 25 U -25 U, those are also going to add to zero. And so we're only left with two terms and we get our simplified expression is UQ -125. And we can't simplify any further than that. If we compare this with our answer choices, we found that after we multiplied our polynomials together and simplified the result, we got U cubed minus 125 which corresponds with answer choice. C 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−1)(x²+x+1)

TextbookSolutions.keyConceptsTitle

Polynomial Multiplication

Horizontal and Vertical Formats

Combining Like Terms

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