Add the following expressions and simplify if possible: x x + 2 + 2 ( x + 2 ) ( x + 3 ) \frac{x}{x+2}+\frac{2}{\left(x+2\right)\left(x+3\right)}

problem asks us to add the following expressions and simplify if possible. I can see I'm adding these two rational expressions. And to add them together, I need to get a common denominator. So what I'm gonna do is find the least common denominator. Now I can see everything is already factored. We have this completely factored, x over x plus two. Everything's factored as this x plus two times x plus three and then this two. So there's nothing else I need to factor out here to simplify or make anything look more straightforward. It's as straightforward as it's gonna get. What I need to do is find my LCD, which is going to be a product of all the unique prime factors. I have an x plus two in this fraction and in the other fraction, but notice I only need to write it once since I'm trying to find the unique prime factor. So we have an x plus two, then we have an x plus three. That's unique and that's the only one. So our LCD is going to be x plus two times x plus three. Now, I already have my least common denominator here, so I don't need to change anything about this second fraction. So the two over x plus two times x plus three, this is going to stay the exact same. The only thing that's going to change is this first fraction, the x over x plus two, because what I need to do is multiply the numerator and denominator by the missing factor. We have the x plus two, so I need to multiply everything by x plus three. So I have an x plus three multiplied on top, and an x plus three multiplied on bottom. Now we're adding these together. So let's go ahead and go through this process here. Well, I can distribute the x into the parentheses. That'll give me x squared plus three x plus two, and then all in the denominator will have what we have right now here, which is x plus two times x plus three. Now at this point, what I can actually do is see if there's any places where I can simplify what we have, because I notice that I end up with a quadratic polynomial in the numerator. It might be tempting to just say we're finished here, and we would be, except we are asked to simplify if possible. And it's possible there are gonna be terms that cancel, so let's give this a try. We have a quadratic polynomial with one as the leading coefficient in front of x squared. So I need a pair of numbers that multiply to give me two and add to give me three. If I think about it, that's going to be one and two. One times two is two, and one plus two is three. So we end up getting x plus one times x plus two in the numerator, and then in the denominator, we have x plus two times x plus three. And I can go ahead and cancel the x plus two, leaving me with x plus one in the numerator and x plus three in the denominator, giving me the solution to part b of this problem.

Add the following expressions and simplify if possible: x x + 2 + 2 ( x + 2 ) ( x + 3 ) \frac{x}{x+2}+\frac{2}{\left(x+2\right)\left(x+3\right)}

1 ( x + 3 ) \frac{1}{\left(x+3\right)}

( x + 1 ) ( x + 4 ) ( x + 2 ) ( x + 3 ) \frac{\left(x+1\right)\left(x+4\right)}{\left(x+2\right)\left(x+3\right)}

1 ( x + 2 ) ( x + 3 ) \frac{1}{\left(x+2\right)\left(x+3\right)}

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Different Denominators

Beginning Algebra

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Different Denominators

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Multiple Choice

Add the following expressions and simplify if possible:
$\frac{x}{x + 2} + \frac{2}{(x + 2) (x + 3)}$

$\frac{1}{(x + 3)}$

$\frac{(x + 1) (x + 4)}{(x + 2) (x + 3)}$

$\frac{x + 1}{x + 3}$

$\frac{1}{(x + 2) (x + 3)}$

Verified step by step guidance

Identify the expressions to be added: $\frac{x}{x+2} + \frac{2}{(x+2)(x+3)} + \frac{1}{x+3} + \frac{(x+1)(x+4)}{(x+2)(x+3)}$.

Find the least common denominator (LCD) for all fractions. Since denominators are $x+2$, $x+3$, and $(x+2)(x+3)$, the LCD is $(x+2)(x+3)$.

Rewrite each fraction with the LCD as the denominator: - $\frac{x}{x+2} = \frac{x(x+3)}{(x+2)(x+3)}$, - $\frac{1}{x+3} = \frac{x+2}{(x+2)(x+3)}$, - The other fractions already have the LCD denominator.

Combine all fractions over the common denominator: $\frac{x(x+3)}{(x+2)(x+3)} + \frac{2}{(x+2)(x+3)} + \frac{x+2}{(x+2)(x+3)} + \frac{(x+1)(x+4)}{(x+2)(x+3)}$.

Add the numerators together and simplify the resulting expression by expanding and combining like terms, then factor if possible to simplify the fraction.

6:23m

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Master Adding and Subtracting Rational Expressions with Unlike Denominators with a bite sized video explanation from Patrick Ford

Struggling with Beginning Algebra?

Add the following expressions and simplify if possible:xx+2+2(x+2)(x+3)\(\frac{x}{x+2}\)+\(\frac{2}{\left(x+2\right)\left(x+3\right)}\)

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Add the following expressions and simplify if possible:
$\frac{x}{x + 2} + \frac{2}{(x + 2) (x + 3)}$