Question

Add the rational expressions to find the sum:  x2−23x−2+x+53x−2 .(1 point)Responsesx3+33x−2Start Fraction x cubed plus 3 over 3 x minus 2 End Fractionx2+x+33x−2Start Fraction x squared plus x plus 3 over 3 x minus 2 End Fractionx2+x+73x−2Start Fraction x squared plus x plus 7 over 3 x minus 2 End Fractionx2+x+36x−4

Answer 1

To add the rational expressions \( \frac{x^2-23x-2}{x+5} + \frac{3x-2}{3} \), we first need a common denominator.

Step 1: Factor where possible

Let's see if we can factor \( x^2 - 23x - 2 \):

The expression \( x^2 - 23x - 2 \) does not factor nicely, so we'll leave it as is.

Next, let's express the second term:

\[ \frac{3x-2}{3} = \frac{3x-2}{3} \]

Step 2: Find a common denominator

The common denominator between \( x + 5 \) and \( 3 \) is \( (x + 5)(3) \).

Step 3: Rewriting each expression with the common denominator

Now, we can rewrite each expression as follows:

1. First term: \[ \frac{x^2 - 23x - 2}{x + 5} = \frac{(x^2 - 23x - 2) \cdot 3}{(x + 5) \cdot 3} = \frac{3(x^2 - 23x - 2)}{3(x + 5)} \]

2. Second term: \[ \frac{3x - 2}{3} = \frac{(3x - 2)(x + 5)}{3(x + 5)} \]

Step 4: Combine the expressions

Now, we can add the two fractions:

\[ \frac{3(x^2 - 23x - 2) + (3x - 2)(x + 5)}{3(x + 5)} \]

Step 5: Expand the numerator

1. Expand \( 3(x^2 - 23x - 2) \): \[ 3x^2 - 69x - 6 \]

2. Now expand \( (3x - 2)(x + 5) \): \[ 3x^2 + 15x - 2x - 10 = 3x^2 + 13x - 10 \]

Step 6: Combine both expansions

Now combine them:

\[ 3x^2 - 69x - 6 + 3x^2 + 13x - 10 = 6x^2 - 56x - 16 \]

Step 7: Final expression

So we have:

\[ \frac{6x^2 - 56x - 16}{3(x + 5)} \]

Now simplifying the numerator \( 6x^2 - 56x - 16 \) gives:

Factor out 2:

\[ = \frac{2(3x^2 - 28x - 8)}{3(x + 5)} \]

Conclusion

Since none of the answers provided in the original question match with this final expression, please verify the inputs of the problem or share the options again.