In Exercises 1–68, factor completely, or state that the polynomia...

Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. (c + d)⁴ − 1

Accepted Answer

Hello, everyone. We are going to perform a complete factorization of the given expression or categorize it as a prime polynomial. The expression we're given is open parentheses. M plus N closed parentheses raised to the fourth power minus 625. Our answer choices are a open square bracket. Open parenthesis, M plus N closed parentheses. Squared plus 25 closed square bracket multiplied by open parenthesis. M plus N plus five closed parentheses, multiplied by open parentheses. M plus N minus five closed parentheses. B open square bracket, open parentheses, M plus N closed parentheses. Squared plus 25 closed square bracket, multiplied by open parentheses. M minus N plus five closed parentheses multiplied by open parentheses. M plus N minus five closed parentheses, C open square bracket, open parentheses, M plus N closed parentheses. Squared plus 25 closed square bracket. Multiplied by open parentheses. M plus N plus five closed parentheses multiplied by open parentheses. M minus N minus five closed parentheses. D the given polynomial is prime. Going back to our original expression. We have open parentheses. M plus N closed parentheses. Raise the fourth power -625. First thing I notice is that if I treat the parentheses as if they are one single term, I have the difference of two squares. So that's what I'm gonna do. I'm going to treat the parentheses. M plus N as if they were one single term, I will not be breaking them apart until probably close to the last step. Recall that when we have the difference of two squares, we are gonna factor one binomial into two binomials. So I'm gonna open a set of square brackets and open a set of parentheses. Actually, I'm gonna open two sets of square brackets just to make sure we're clear on which thing goes where. So first to find the first term in my new binomial, I take the square root of the first term in the original binomial. So the square root of in parentheses M plus N raised to the fourth power, the square root of that is gonna be parentheses. M plus N closed parentheses squared. So that's the first term in my new binomial. So open parentheses, M plus N, close parentheses squared. To find the second term in my new binomial, I need to take the square root of the second term in my original binomial. So the square root of 625, which is 25. So that is the second term in both sets of parentheses. Oh, I'm sorry, in both sets of square brackets and one needs to have a plus sign while the other has a minus sign. So right now, our factorization is open square bracket, open parentheses, M plus N closed parentheses, squared plus 25 closed square bracket, multiplied by open square bracket. Open parentheses, M plus N closed parentheses, squared minus 25 closed square bracket. To do a complete factorization. I need to see if either these binomials can be factored further. The first binomial, the parentheses M plus N squared plus 25 cannot be factored any farther because it's not the difference of squares. It's the sum and we can't factor the sum. However, the second set of brackets where we have in parentheses, M plus N squared minus 25 is again the difference of squares. So I'm gonna rewrite my first set of square brackets. So open square bracket, open parentheses, M plus N closed parentheses, squared plus 25 closed square bracket. And then I'm gonna open, you said to parentheses. And I'm gonna use those as my new binomials. The factor the parentheses M plus N squared minus 25. So again, we have the difference of two squares. So the first term in my new parentheses comes from the square root of the first term in the binomial of the difference of two squares. So the square root of M plus N squared is M plus N. And because there's no sign in front of them in these parentheses, I can drop the parentheses that I was keeping around. M plus N. So in our parentheses, we have M plus N, we need to find the last term for this new binomial. And that's gonna be the square root of 25, which is five. We need to put signs in between the N and the five. So we need one set of parentheses to have a plus sign and the other one to have a minus sign. So our factorization is open square bracket multiplied by open parentheses. M plus N closed parentheses, squared plus 25 closed square bracket, multiplied by open parentheses. M plus N plus five closed parentheses multiplied by open parentheses. M plus N minus five closed parentheses. There is nothing else here that I can factor. So this is my solution and it matches answer choice a have a nice day.

In Exercises 1–68, factor completely, or state that the polynomial is prime. (c + d)⁴ − 1

Key Concepts

Factoring Polynomials

Difference of Squares

Binomial Theorem

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