In Exercises 69–80, factor completely.

x⁴ − 5x²y² + y⁴

Question

In Exercises 69–80, factor completely.

x⁴ − 5x²y² + y⁴

Accepted Answer

Hello, everyone. We are going to completely factor the expression A raised to the fourth power minus 10 A squared B squared plus nine B raised to the fourth power. Our answer choices are A, the parentheses A squared plus three B multiplied by the parentheses, A minus three B multiplied by the parentheses A plus B B, the parentheses A plus three B multiplied by the parentheses. A minus three B multiplied by the parentheses. A plus B multiplied by the parentheses, A minus B C, the parentheses A squared plus three B multiplied by the parentheses A squared minus three B D, the parentheses A squared plus three B multiplied by the parentheses. A minus B multiplied by the parentheses. A plus three B wanna rewrite my original expression. A raise the fourth power minus 10 A squared B squared plus nine B raised the fourth power. I recall that when I factor an expression, the first thing I wanna do is look for any greatest common factors. So is there any common factors between these three terms? And there is not. So next, I noticed that A of the fourth power and 9 b to the fourth power are perfect squares. So I wanna see if this is a perfect square trinomial, if it is the middle term should be the square of the first term multiplied by the square of the last term multiplied by two. And so this would be like a three times two, this would have to be a six. So it's not a perfect square trinomial, which means I'm going back to kind of reverse foil to find out what this factors into. So I opened two empty sets of parentheses and now to find the first term in each set of parentheses, I want two values that multiply to a raise the 4th power and that's gonna be a squared times A squared. Next, I'm gonna think about what signs I'm gonna have in this binomial. I notice that my last term is positive and I know that my last terms from the binomial need to multiply to my last term of positive 9B raised to the 4th power but need to add to negative 10 A squared B squared. So if they need to multiply to a positive and add to a negative, this means both signs are minus signs because a negative times a negative will give me the positive and add to the negative. Next, I'm looking for two values that multiply to nine but add to 10. So first I'm gonna check one and nine. Well, one in nine, definitely multiply to nine and add to 10. So I'm gonna put a one and then nine as my terms in my binomial. But I can't forget that this is also be raised to the 4th power. So I need to multiply two things together to get b raise the fourth power. So that will be B squared and B squared. So at this step in factorization, I have open parentheses. A squared minus B squared, closed parentheses, multiplied by open parentheses. A squared minus nine B squared, closed parentheses. I wanna see if this can be factored any further. And it looks like both sets of parentheses have the difference of two squares. So being that both sets of parentheses are the difference of two squares, I'm going to use that format to factor this further. So I'm gonna start with the A squared minus B squared. I recall that when I factor the difference of two squares, I end up with two binomials. So I've opened two empty sets of parentheses so I can fill them. The first term in my new binomials comes from the square root of the first term in the difference of two squares. So the square root of A squared, which is a, the first term in each, the second term comes from the square root of the last term. So the square root of B squared which is B and one needs a plus sign, one needs a minus sign. So the factorization of this piece we're not done yet is the parentheses A plus B multiplied by the parentheses A minus B. Next, I need to do the difference of two squares for the parentheses. A squared minus nine B squared. So again, I'm gonna open two empty sets of parentheses. The first term in these parentheses is gonna come from the square root of the first term in A squared minus nine B squared. So the squared of A squared is A. So that will be the first term in each binomial. The square root of nine B squared is three B squared, I'm sorry. Three B. So three B is the second term in both. And again, one needs a plus sign, one needs a minus sign. So one plus sign, one minus sign at this point. My factorization is the parentheses A plus B multiplied by the parentheses. A minus B multiplied by the parentheses. A plus three B multiplied by the parentheses A minus three B. This is my answer. However, at the moment, it does not match any of our answer choices. This is because if you recall multiplication can be done in any order. So as long as we have all the same factors, the order doesn't matter. So this matches answer choice B as it has all of the same factors, they just put them in a slightly different order.

In Exercises 69–80, factor completely. x⁴ − 5x²y² + y⁴

Key Concepts

Factoring Polynomials

Difference of Squares

Quadratic Form

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