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What is the LCD in its factored form of the following rational expression subtraction?

x2+2xx2+7x−8−xx−1

(1 point)
Responses

x(x+2)
x left parenthesis x plus 2 right parenthesis

(x−1)2(x+8)
left parenthesis x minus 1 right parenthesis squared left parenthesis x plus 8 right parenthesis

(x−1)(x+8)
left parenthesis x minus 1 right parenthesis left parenthesis x plus 8 right parenthesis

(x−1)(x2+7x−8)

1 answer

To find the least common denominator (LCD) in its factored form of the given rational expression subtraction:

\[ \frac{x^2 + 2x}{x^2 + 7x - 8} - \frac{x}{x - 1} \]

we will first factor each denominator.

  1. Factor the first denominator \(x^2 + 7x - 8\): We need to find two numbers that multiply to \(-8\) and add to \(7\). These numbers are \(8\) and \(-1\). Therefore:

    \[ x^2 + 7x - 8 = (x + 8)(x - 1) \]

  2. Factor the second denominator \(x - 1\): This is already factored.

Now that we have the denominators factored, the denominators are:

  • The first denominator: \((x + 8)(x - 1)\)
  • The second denominator: \((x - 1)\)

To determine the least common denominator (LCD), we take the highest power of each factor present in the denominators:

  • From \((x + 8)\), we take \(x + 8\).
  • From \((x - 1)\), we take \((x - 1)\) (since it appears in both).

Thus, the LCD is:

\[ \text{LCD} = (x + 8)(x - 1) \]

Therefore, the correct response, expressed in its factored form, is:

\[ (x - 1)(x + 8) \]

So the answer is: (x−1)(x+8).