Exercises 108–110 will help you prepare for the material covered ...

Question

Exercises 108–110 will help you prepare for the material covered in the next section.

Simplify and express the polynomial in standard form:
3x(x² + 4x + 5) + 7(x² + 4x + 5)

Accepted Answer

Hey, everyone in this problem, we're asked to represent the given polynomial in standard form. After simplification, the polynomial we're given is five X multiplied by the polynomial two X squared plus five X plus seven plus three, multiplied by the polynomial X squared plus eight, X plus one. We're given four answer choices. Option A 13 X cubed plus 28 X squared plus 59 X plus three. Option B 10 X cubed plus 28 X squared plus 59 X plus three. Option C 10 X cubed plus X squared plus 35 X plus three. And option D 10 X cubed plus 28 X squared plus 24 X plus three. We're gonna start by writing the expression we were given. So we have 5 x multiply by two X squared plus five, X plus seven plus three, multiplied by X squared plus eight X plus one. We're gonna start our simplification by distributing this five X term to each term in the polynomial that it's being multiplied by. Yes, we're gonna take the five X and multiply it to two X squared. We're gonna multiply it to 5 x and then we're gonna multiply it to seven. In order to expand these terms, we're gonna do the same with the second half of the equation. We have three multiplied by the polynomial X squared plus eight X plus one. So we need to distribute that three to every term. So we're gonna multiply it to the X squared then to the 8 x then to the one, when we do that, we get the followed, we have five X multiplied by two X squared. We're gonna add our next term is five X multiplied by five X. We add our next term 5 x multiplied by seven. And that's it. For the first part of this equation or expression, we move to the second part of our expression. We add, we have three multiplied by X squared. We add our next term three multiplied by 8 x in our final term. Three multiplied by one. And those are all of the terms in our polynomial that we have. Now, we can get to simplifying each of these terms. Starting with our first term. We have five X multiplied by two X squared. The coefficient we have is five multiplied by two. So we get a coefficient of 10 and we have X multiplied by X squared. This gives us X to the exponent one plus two can recall that if you have two things multiplied together with the same base, you can combine them into a single term where the exponents are added, moving to our next term five, X multiplied by five, X five, multiplied by five, gives us a coefficient of 25. And we have X multiplied by X. We can write this as X to the exponent one plus one. We move to our next term 5 x multiplied by seven. So give us 35 x. Our next term three multiplied by X squared. This gives us three, X squared, three multiplied by 8 x is this 24 x in our final term, three multiplied by one, gives us three. And so we have all six of those terms. And we're getting closer. Let's take our next step to be simplifying these exponents that we have. So we have 10 to the ex or sorry, 10 multiplied by X to the exponent one plus two. So we get 10 X to the exponent three. We add, we have 25 X to the exponent one plus one gives us X squared. And the remainder of the terms we're gonna leave alone for this step. So we have plus 35 x plus 3 x squared Plus 24 x plus 3. Now I can simplify a little further. I notice that we have 2 x squared terms, we have two X terms. So we can combine these terms to simplify starting with our X cube term, we only have one X cube term. So we have 10 execute, moving to our X squared terms. We have 25 X squared. So we have three X square, OK, 25 plus three gives us 28 X squared. Next, we move to our X terms. We have 35 X and we have 24 X, 35 X plus 24 X gives us 59 X and then we have our constant plus three at the end. And now this polynomial is simplified. It's in standard form. OK? Where we have our X cube term, our X squared term, our X term and our constant term. And that is the result we were looking for if we compare this to our answer choices and we've found our polynomial in standard form after simplification is 10 X cubed plus 28 X squared plus 59 X plus three, which corresponds with answer choice. B Thanks everyone for watching. I hope this video helped see you in the next one.

Exercises 108–110 will help you prepare for the material covered in the next section. Simplify and express the polynomial in standard form: 3x(x² + 4x + 5) + 7(x² + 4x + 5)

Key Concepts

Polynomials

Distributive Property

Standard Form of a Polynomial

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