Work each problem. Which of the following is the correct compl...

Question

Work each problem. Which of the following is the correct complete factorization of $x^{4} - 1$ ?

A. $(x^{2} - 1) (x^{2} + 1)$

B. $(x^{2} + 1) (x + 1) (x - 1)$

C. $open paren x squared minus 1 close paren squared$

D. $(x - 1)^{2} (x + 1)^{2}$

Accepted Answer

Hey everyone here we are asked to show the complete factory ization of 16 M two. The fourth power minus 81. Now our first step here in factoring our expression will be to rewrite our two terms as squares. So beginning with our first term which is again 16 M two. The fourth power. Well writing it as its square, we can take the square root of this term which gives us four M squared. And then we just need to square this quantity. So it is equivalent to our first term. And now we subtract our second term which is 81. Well the square root of 81 is just nine. So we have the second term rewritten as nine squared. So now here with our rewritten expression, we can clearly see that we have the difference of two squares. So we need to rick call the difference of squares formula which tells us that x squared minus Y squared is equal to the quantity of X plus y times the quantity of x minus y. So in this case we can identify our x and y variables in relation to our expression here. So we see that X. Our first term is going to be four M squared and then our second term, the term we're subtracting here is our Y. So y is going to be nine. And now we can go ahead and use this formula to continue to factor out our expression. So again we have the quantity of four M squared raised to the second power and then we have minus the quantity of nine squared. And now rewriting using our formula, we have the quantity of four M squared plus nine times the quantity of four M squared minus nine. And now we are not done factoring quite yet. We can take a look at our second set of parentheses here and we see that we actually have the difference of squares once again. So we can go ahead and start by rewriting our expression here, rewriting the terms in our expression as squares. So doing this, we have the quantity of four M squared plus nine times the quantity of two M squared minus three squared. And with this we can see using the difference of squares formula that our X. Is now going to be two M. And then our Y is now going to be three. So again we are factoring out our expression of the quantity of four M squared plus nine times the quantity of the quantity of two M squared minus three squared. Which again using the formula we can factor out this expression as the quantity of four M squared plus nine times the quantity of two M plus three Times The Quantity of two M -3. And here we can see that we have now fully factored out our original expression and therefore we are done here solving this problem and we are left with answer choice D. Thanks for watching. I hope you found this video helpful and I'll see you again next time

Work each problem. Which of the following is the correct complete factorization of $x^{4} - 1$ ?
A. $(x^{2} - 1) (x^{2} + 1)$
B. $(x^{2} + 1) (x + 1) (x - 1)$
C. $open paren x squared minus 1 close paren squared$
D. $(x - 1)^{2} (x + 1)^{2}$

Key Concepts

Difference of Squares

Factoring Quadratic Expressions

Complete Factorization

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