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

Question

Add or subtract, as indicated. See Example 4. 4/(x+1) + 1/(x² - x + 1) - 12/(x³ + 1)

Accepted Answer

Hello, everyone. We are asked for the following rational expression to perform the indicated operations and simplify. If necessary. We are given three divided by the parentheses B plus three plus 16 divided by the parentheses. B squared minus three B plus nine minus 81 divided by the parentheses. B cubed plus 27. Our answer choices are A, the parentheses two minus three B divided by the parentheses B minus nine B, the parentheses two minus three B divided by the parentheses, B plus nine C, the parentheses three B minus two divided by the parentheses B squared plus three B minus nine D, the parentheses three B minus two divided by the parentheses B squared minus three B plus nine. First, I'm gonna rewrite this so that all of my division are fractions. So I have three divided by B plus three plus 16, divided by B squared minus three B plus nine minus divided by B cubed plus 27. So in problems like this, the first thing I do when I see a minus sign is I change it to A plus sign and change the numerator that follows it. So I'm gonna take this minus sign, make it a plus sign and make 81 negative. I have just found that I lose track of minus signs a lot faster than I lose track of plus signs for some reason. So it's a helpful hint. Next, I know in order to combine these, I need to have a common denominator. So I might need to factor all of these to find out what that is. So I'm gonna start my first fraction three, divided by B plus three can't be factored any further. My second fraction 16, we have it divided by B squared minus three, B plus nine. And when I go to factor that, I recognize that nothing multiplies to a positive nine and adds to a negative three. So I actually can't factor that any further. So let's see what, how this works out. And then we're gonna add the last fraction. So our numerator is negative 81. I can factor B cubed plus 27. However, because recall that if we have the sum of cubes like X squared plus Y squared, that is the same as saying we have X plus Y multiplied by X squared minus XY plus Y squared. So if we follow that format, we have B cubed plus 27. So this means X cubed is equal to B cubed. So then X would have to equal B and Y cubed would be equal to 27 which means why is the cube root of 27 which is three. So if I plug that into the factorization up here, I have X plus Y. So B plus three multiplied by X squared. So B squared minus X multiplied by Y so B multiplied by three, which is three B plus Y squared. So three squared which is nine. So this factorization can be put down as our denominator instead of B cubed plus 27 we can rate B plus three in parentheses, multiplied by B squared minus three B plus nine. Notice how the denominator for negative 81 is both factors from the other two denominators that is great for us. This will make this a lot easier because we now know our common denominator. It has to be the parentheses. B plus three multiplied by the parentheses B squared minus three B plus nine. So we're gonna create three fractions that have that denominator and we will see what happens when we combine them. So I've set the denominators of all three of them as the parentheses B plus three multiplied by the parentheses B squared minus three B plus nine. So if you recall when you had traditional fractions, if you multiplied the bottom by four, you'd multiply the numerator by four. Same here, except where it's a little more complicated than four. So in our first fraction, we multiplied our denominator by the parentheses B squared minus three B plus nine. So we need to multiply our numerator, which was a three by B squared minus three B plus nine, which will be in parentheses. Our second fraction, we had the B squared minus three B plus nine. We multiplied it by B plus three. So we need to multiply 16 by B plus three. In their last fraction. We already had all those pieces in our denominator. So that stays minus 81. Now, I'm gonna distribute to simplify my numerators. So when I do, I have three B squared minus nine B plus 27 and I'm gonna write this as one fraction. So every numerator is gonna be added together over my denominator of in parentheses. B plus three multiplied by the parentheses B squared minus three B plus nine. So I've written down the info from the first numerator. The second num reader, we distribute 16 into the parentheses. So we get plus 16 B plus 48 and then our last n reader was minus 81. All right. So now let's combine our like terms in our numerator. So three B squared, there's no other B squared. So that's that we have a minus or a negative nine B and A positive 16 B. So that'll combine to a positive seven B and our constants, we have 27 48 and negative which will be a negative six or minus six. And all of that is over a common denominator of B plus three multiplied by the parentheses B squared minus three B plus nine. So now I wanna see if I can factor my numerator. So it will cancel with any piece of my denominator. So we're gonna factor our numerator into two binomials and our denominator will stay the same for now. So we have the parentheses and our denominator B plus three multiplied by B squared minus three B plus nine closed parentheses. So in my numerator, my first term in my new binomials, one will have to be three B and one will be B to multiply back to three B squared. And then for the second term in those new binomials, I know I have to have opposite signs because it multiplies to a negative six, but it needs to add to a positive seven. So I'm gonna go with, we'll have three B minus two, multiplied by the parentheses B plus three. And this would make sense even double checking it inner and outer terms, they do add up to a positive seven B and I now can reduce the factor B plus three in my numerator with the factor B plus three in my denominator. And that leaves me with three B minus two divided by B squared minus three B plus nine as my final fraction as simplified as it gets. And I'm going to look at my answer choices and it matches answer choice D. So we have the parentheses three B minus two divided by the parentheses B squared minus three B plus nine. Have a nice day.

Add or subtract, as indicated. See Example 4. 4/(x+1) + 1/(x² - x + 1) - 12/(x³ + 1)

Key Concepts

Polynomial Addition and Subtraction

Finding a Common Denominator

Factoring Polynomials

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