In Exercises 65–74, factor by grouping to obtain the difference o...

Question

In Exercises 65–74, factor by grouping to obtain the difference of two squares.

9x^2 − 30x + 25 − 36x^4

Accepted Answer

Hello, everyone. We are gonna completely factorize the following difference of two squares by first grouping the polynomial we are given is 25 M squared minus 20 M plus four minus 16 M raised to the fourth power. Our answer choices are a negative multiplied by the parentheses. Four M squared minus five M plus two closed parentheses, multiplied by open parentheses. Four M squared plus five M minus two closed parentheses. B open parentheses, four M squared minus five M plus two closed parentheses, multiplied by open parentheses. Four M squared plus five M minus two closed parentheses. See negative open parentheses. Four M squared minus five M plus four, closed parentheses, multiplied by open parentheses. Four M squared plus five M minus four, closed parentheses. D open parentheses, four M squared minus five M plus four, closed parentheses, multiplied by open parentheses. Four M squared plus five M minus four, closed parentheses. I'm going to rewrite my original polynomial before I do my grouping. So my original polynomial is 25 M squared minus 20 M Plus 4 -16 M raised to the 4th power. I recall that most of the time when I am factoring by grouping, I am grouping by sets of two terms. In this case, if I want this to be the difference of two squares, I'm not gonna factor this by grouping the first two terms together and the last two terms together. But instead by grouping the first three terms together. So I put parentheses around 25 M squared minus 20 M plus four because I recognize that this is a perfect square trinomial recall that in a perfect square trinomial, you're gonna end up with a binomial in a set of parentheses. And the whole parentheses will be squared. We can check to see if this is a perfect square trinomial by taking the square of the last term, the square of the first term multiplying them together and then doubling it. So just to double check the square of the first term would be five M where the last term would be 25 M multiplied by two would be 10 M doubled, gives me 20 M, which is my middle term. So now I can factor this the binomial that we will square to get this perfect square trinomial. The first term is the square root of the first term from the trinomial. So the square root of 25 M squared is five M. The last term is the square of the last term in the trinomial. So the squared of four is two and the sign in between them is the sign on the middle term from the trinomial. In this case, it's a minus sign. So the first three terms of our original polynomial will factor two open parentheses, 5 M -2 closed parentheses squared. And now that's still gonna be -16 M raised to the 4th power. But now we have our difference of two squares. We have our difference of two squares because we now have two perfect squares. So we're gonna factor it as such. I'm gonna keep the parentheses five M minus two group together until the very end recall that when we do the difference of two squares, we start with two grouping symbols. So we can have two binomials. I'm gonna use square brackets in this case because I know I'm working with a set of parentheses within them. So my Bino meals are gonna be in two sets of square brackets that are correct. Currently, next to each other recall that when you factor the difference of two squares, the first term in your new binomials comes from the square root of the first term in the difference of two squares. So this would be the square root of the parentheses, five M minus two squared and the square root of that is 5M -2. And I'm keeping that in parentheses just for the moment. So I can group them together. But that's gonna be treated as my first term in each new set of binomials. The second term comes from the square root of the second term in the difference of two squares. So the square root of 16 M rays to the fourth power, which will be four M squared. That's the second term in both binomials, one needs to have a plus sign in between them and one needs to have a minus sign. So right now without rearranging anything, I have open square bracket, open parentheses, five M minus two closed parentheses plus four M squared, closed square bracket, multiplied by open square bracket, open parentheses, five M minus two, closed parentheses minus four M squared, closed square bracket. Now, I see that there's nothing that needs to be distributed into the parentheses. 5 M -2. So I can drop those parentheses and I'm gonna rearrange all my terms so that inside of the grouping symbol, the exponents decrease. So I'm gonna now work with parentheses. And in the first set, I have a four M squared which I'm gonna put first, then plus five M minus two. And you close those parentheses, open the next set of parentheses and I have negative four M squared plus five M -2 closed parentheses. So we're almost done if you notice there are not answer choices that have a negative sign within a grouping symbol on the M squared term. This means they want us to factor out the negative one from that term. So I'm gonna do that. My second set of parentheses has negative four M squared plus five M minus two. And I'm gonna factor out a negative one. It's not gonna change what I have in my first set of parentheses, which is four M squared plus five M minus two. But in my second set of parentheses, it will change all of the signs. So we had a negative four M squared. Now it's positive four M squared. We had a positive five M. Now it's minus five M. We had M negative two. Now we have plus two. So I now have negative one, multiplied by open parentheses. Four M squared plus five M minus two closed parentheses, multiplied by open parentheses. Four M squared minus five M plus two closed parentheses. Though my sets of parentheses are in the opposite order. That's OK. In multiplication, this is our answer and it matches answer choice. A recall that in multiplication, our factors can go in either order. So it is all right that they have it as negative multiplied by open parentheses. Four M squared minus five M plus two, closed parentheses multiplied by open parentheses. Four M squared plus five M minus two, closed parentheses have a nice day.

In Exercises 65–74, factor by grouping to obtain the difference of two squares. 9x^2 − 30x + 25 − 36x^4

Key Concepts

Factoring by Grouping

Difference of Squares

Polynomial Expressions

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