Question

question #8
camryn is subtracting the following rational expressions: $\frac{4}{2x - 5}-\frac{2x}{x + 1}$. camryns work is shown.
did camryn make a mistake? if so, in which step was the mistake made? what was the mistake?
step 1: lcd is $(2x - 5)(x + 1)$
step 2: $\frac{4(x + 1)}{(2x - 5)(x + 1)}-\frac{2x(2x - 5)}{(2x - 5)(x + 1)}$
step 3: $\frac{4x + 4}{(2x - 5)(x + 1)}-\frac{4x^{2}-10x}{(2x - 5)(x + 1)}$
step 4: $\frac{-4x^{2}+14x + 4}{(2x - 5)(x + 1)}$
step 5: $\frac{-2(2x^{2}-7x - 2)}{(2x - 5)(x + 1)}$
step 2: camryn multiplied each rational expression by the incorrect factor to create common denominators.
no errors were made. the work is correct.
step 3: camryn simplified the numerators from step 2 incorrectly.
step 5: camryn factored the simplified numerator incorrectly.

Explanation:

Step1: Check Step 1

Finding the least - common denominator (LCD) of two fractions $\frac{4}{2x - 5}$ and $\frac{2x}{x + 1}$. The LCD of two non - zero polynomials $2x-5$ and $x + 1$ is indeed $(2x - 5)(x + 1)$. So Step 1 is correct.

Step2: Check Step 2

To get a common denominator, we multiply the numerator and denominator of $\frac{4}{2x - 5}$ by $x + 1$ and $\frac{2x}{x + 1}$ by $2x - 5$. We get $\frac{4(x + 1)}{(2x - 5)(x + 1)}-\frac{2x(2x - 5)}{(2x - 5)(x + 1)}$. Step 2 is correct.

Step3: Check Step 3

Expand the numerators: $4(x + 1)=4x+4$ and $2x(2x - 5)=4x^{2}-10x$. So the fractions are $\frac{4x + 4}{(2x - 5)(x + 1)}-\frac{4x^{2}-10x}{(2x - 5)(x + 1)}$. Step 3 is correct.

Step4: Check Step 4

Subtract the numerators: $(4x + 4)-(4x^{2}-10x)=4x + 4-4x^{2}+10x=-4x^{2}+14x + 4$. So $\frac{-4x^{2}+14x + 4}{(2x - 5)(x + 1)}$. Step 4 is correct.

Step5: Check Step 5

Factor out $- 2$ from $-4x^{2}+14x + 4$. We have $-4x^{2}+14x + 4=-2(2x^{2}-7x - 2)$. Step 5 is correct.

Answer:

No errors were made. The work is correct.