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

this problem, we are asked to add the following expressions and simplify if possible. So I have x over x squared plus three x, plus x over x plus three. So I'm trying to add two rational expressions. Now a good step to start, whenever you're really dealing with these rational expressions, is to just factor everything that can be factored. Now the the reason that we should factor first is because this will clearly show us any types of things we have in common with the numerators and denominators, or in general finding the LCD. So let's go ahead and do this. Now, first off, I'm gonna factor this denominator, x squared plus three x. I can pull an x out of each of these terms, giving me x times x plus three. So we have everything factored for the left rational expression, and everything on the right is already factored as well. So we have three over x plus three, there's nothing more to do from here. So now that I have the factored rational expressions, what I need to do is add these together by getting a common denominator. And I can get a common denominator by finding the LCD. Now the least common denominator is going to be a product of all the unique prime factors that we have. I can see we have an x just by itself, and I also see we have this x plus three. So these are all the unique prime factors. Now notice that we have the LCD right here, so nothing is going to change. We'll have x over x times x plus three. However, what I can see is that we also have this rational expression where I can multiply the numerator and denominator by the missing factor of x. So we'll keep this three over x plus three. But what I can see is that I can multiply this by x and then in the numerator and the denominator. So whatever I do to the denominator, I also have to do to the numerator. But notice by doing this, we get a common denominator. So that means that in the numerator, we're going to have x plus three x, which gives us four x, and then in the denominator we'll have x times x plus three. Now notice how we have an x in common as well, so I can cancel this x. So that's going to give me four over x plus three. And this right here would be the solution to the problem. Now something else that I'm going to mention here is you could have actually done this step differently. If you wanted to, you could have just immediately canceled these x's right about here since we have an x in the numerator and denominator of this first rational expression, and that would give you one over x plus three plus three over x plus three, which would all end up becoming four over x plus three. And so that step would be a little bit faster, but it might also feel a little bit less natural since we're used to finding the LCD first and then getting common denominators and working from there to simplify. So either one of these steps would have been perfectly fine to get to this answer, but as long as you're able to get four over x plus three, this is the strategy for solving the problem. Hope you found this video helpful.

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

4 x 2 + 3 x \frac{4}{x^2+3x} x 2 + 3 x 4

4 x x + 3 \frac{4x}{x+3} x + 3 4 x

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Different Denominators

Beginning & Intermediate 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:
$\frac{x}{x^{2} + 3 x} + \frac{3}{x + 3}$

$\frac{4}{x^2+3x}$

$\frac{4x}{x+3}$

$\frac{4}{x+3}$

$\frac{4}{x}$

Verified step by step guidance

Start by examining the given expression: $\frac{x}{x^2 + 3x} + \frac{3}{x + 3}$. Notice that the denominator $x^2 + 3x$ can be factored.

Factor the quadratic denominator: $x^2 + 3x = x(x + 3)$. Rewrite the first fraction as $\frac{x}{x(x + 3)}$.

Simplify the first fraction by canceling the common factor $x$ in numerator and denominator, resulting in $\frac{1}{x + 3}$ (assuming $x \neq 0$).

Now the expression is $\frac{1}{x + 3} + \frac{3}{x + 3}$. Since both fractions have the same denominator, combine the numerators: $\frac{1 + 3}{x + 3}$.

Add the numerators to get $\frac{4}{x + 3}$. This is the simplified form of the original expression.

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Add the following expressions and simplify if possible:xx2+3x+3x+3\frac{x}{x^2+3x}+\frac{3}{x+3}

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