Add or subtract, as indicated. See Example 4. (3x)/(x² + x − 1...

Question

Add or subtract, as indicated. See Example 4. (3x)/(x² + x − 12) − x/(x² − 16) + x

Accepted Answer

Hello, everyone. We are asked for the following rational expression to perform the indicated operations and simplify. If necessary. We are given four M divided by the parentheses. M squared plus four M minus 12. Close parentheses minus M divided by the parentheses. M squared minus 36. Our answer choices are a open parentheses. Three M squared minus 22 M closed parentheses divided by open bracket. The parentheses M plus six multiplied by the parentheses. M minus six, multiplied by the parentheses. M minus two close square bracket. B open parentheses. Three M squared plus 22. M closed parentheses divided by open square bracket. The parentheses M plus six multiplied by the parentheses. M minus six, multiplied by the parentheses. M plus two closed square bracket. C the parentheses three M minus 22 divided by the parentheses. M minus two D, the parentheses three M plus 22 divided by the parentheses. M plus two. I'm gonna rewrite this and I'm gonna write my division as fractions. So I have four M divided by my denominator is gonna be M squared plus four M minus 12. And we're gonna subtract M divided by M squared minus 36. So I need to create a common denominator. And in order to do that, I will have to factor the denominators I have currently. But I think the first thing I'm gonna do is I'm gonna take this middle minus sign. I'm gonna make it a plus sign and change the sign of the new breeder that follows it. This way. I have no chance of accidentally losing the minus sign in the middle. I've already taken care of it from there. I need to factor my denominators. So my N readers are gonna stay the same. So we'll have four M and then plus a negative M. So the denominator of the first fraction, we have M squared plus four M minus 12. So I know that I will have an M as my first term in each binomial. And then thinking of values that will multiply to a negative 12, but add to a positive four, I'm gonna have a positive six and a negative two. So our denominator there now is the parentheses M plus six multiplied by the parentheses M minus two. For my second fraction I have M squared minus which is the difference of two squares. So I recall when I factor the difference of two squares, I take the square to the first term for the first term in my binomials, the square root of the second term in the original binomial as the second term. So both of them will be six and one will have a plus sign between terms, one will have a minus. So my denominator there is now the parentheses M plus six multiplied by the parentheses M minus six. Next, here's where I'm gonna create my common denominator. So I'm gonna set two fraction bars. So we're gonna create a common denominator. So when I'm creating my common denominator I start with, is there a factor they have in common. Both sets of denominators have M plus six. So I'm writing that down as part of my common denominator. And I put a little checkmark under it just to show myself. I used it. Now, I'm gonna need both pieces of the denominators that aren't common also. So I'm gonna need the parentheses M minus two and the parentheses M minus six. That way I encompass all of the factors of both fractions in my common denominator. So my common denominator is the parentheses M plus six, multiplied by the parentheses, M minus two, multiplied by the parentheses M minus six. So now think about if we had just our standard numbers as a fraction, what we multiplied our denominator by, we have to multiply our numerator by. So comparing step two and step three, my first fraction, my denominator got multiplied by M minus six. So my numerator of four M will be multiplied by the parentheses M minus six. In the second one, the element that was added when we did the common denominator was we multiplied by M minus two. So my numerator of negative M will also be multiplied by M minus two. So now I'm going to use the distributive property and simplify my numerators. So then I can combine to one fraction. So we are going to simplify the name readers and this will take the distributive property. So four M multiplied by M is four M squared, four M multiplied by negative six is negative 24 M over our denominator of M plus six multiplied by M minus two, multiplied by M minus six. And then my second fraction distributing the negative M in negative M multiplied by M is negative M squared. Negative M multiplied by negative two is plus two M. And we still have our common denominator of the parentheses. M plus six multiplied by the parentheses. M minus two, multiplied by the parentheses M minus six. I'm now gonna combine this into one fraction. So I'm gonna combine my like terms in my numerator as I write it over one single denominator, which will be this long lovely denominator. So the parentheses M plus six multiplied by the parentheses M minus two, multiplied by the parentheses and minus six is our denominator. My numerator, I have four M squared and negative M squared which will combine to a positive three M squared. I have negative 24 M and a positive two M which will combine to negative 22 M So this is our final fraction. There is nothing else that factoring will do for us. Here. I could factor an M out of both terms in my enumerator, but it wouldn't reduce. So I'm not going to bother. So our final fraction is three M squared minus 22 M divided by the parentheses. M plus six, multiplied by M minus two, multiplied by the parentheses. M minus six. Looking at my answer choices, it looks like they wrote the order of the denominator slightly differently. So to match them, this would be rewritten as parentheses. Three M squared minus 22 M which is our denominate or sorry, our numerator divided by open square bracket, the parentheses M plus six multiplied by the parentheses. M minus six multiplied by the parentheses M minus two close square bracket. And recall that in our denominator order doesn't matter as long as they're all there because they're factors and in multiplication order won't make a difference. And now that we've written it in a form that isn't directly a fraction more as a division sentence, it matches answer choice. A have a nice day.

Add or subtract, as indicated. See Example 4. (3x)/(x² + x − 12) − x/(x² − 16) + x

Key Concepts

Factoring Quadratic Expressions

Finding a Common Denominator

Adding and Subtracting Rational Expressions

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