Question

Jermey subtracted the following rational expressions. his teacher told him that his answer was incorrect. explain jeremys error in the simplification process and provide the correct results.

2x^2-10\x-5 - x^2+15\x-5 = x^2+5\x-5

Answer 1

This is likely what Jeremy did

`\frac{2\text{x}^2-10}{\text{x}-5}-\frac{\text{x}^2+15}{\text{x}-5}\\\\\frac{2\text{x}^2-10-\text{x}^2+15}{\text{x}-5}\\\\\frac{\text{x}^2+5}{\text{x}-5}\\\\`

The error happens in line 2

This is what his steps should be

`\frac{2\text{x}^2-10}{\text{x}-5}-\frac{\text{x}^2+15}{\text{x}-5}\\\\\frac{2\text{x}^2-10-(\text{x}^2+15)}{\text{x}-5}\\\\\frac{2\text{x}^2-10-\text{x}^2-15}{\text{x}-5}\\\\\frac{\text{x}^2-25}{\text{x}-5}\\\\\frac{(\text{x}-5)(\text{x}+5)}{\text{x}-5}\\\\\text{x}+5`

On the 2nd step, we subtract all of (x^2+15) and not just the x^2 part. The negative distributes to each term in step 3. Then we combine like terms, factor and cancel out the (x-5) terms.

Therefore,
`\frac{2\text{x}^2-10}{\text{x}-5}-\frac{\text{x}^2+15}{\text{x}-5}=\text{x}+5` is an identity as long as
`\text{x} \\e 5` to avoid a division by zero error.