Divide. start fraction start fraction x squared plus 2 x plus 1 over x minus 2 end fraction over start fraction x squared minus 1 over x squared minus 4 end fraction end fraction
A. Start Fraction left-parenthesis lower x plus 1 right-parenthesis left-parenthesis lower x plus 2 right-parenthesis over lower x minus 1 End Fraction
B. Start Fraction left-parenthesis lower x minus 1 right-parenthesis left-parenthesis lower x minus 2 right-parenthesis over lower x plus 1 End Fraction
C. Start Fraction left-parenthesis lower x minus 1 right-parenthesis left-parenthesis lower x plus 2 right-parenthesis over lower x plus 1 End Fraction
D. Start Fraction left-parenthesis lower x plus 1 right-parenthesis left-parenthesis lower x minus 2 right-parenthesis over lower x minus 1 End Fraction
First, factor the numerator and denominator:
Numerator: x^2 + 2x + 1 = (x + 1)^2
Denominator: x^2 - 1 = (x + 1)(x - 1)
Numerator of whole fraction: (x + 1)^2 / (x - 2)
Denominator of whole fraction: (x + 1)(x - 1) / (x + 2)(x - 2)
To divide by a fraction, you invert and multiply. Therefore:
(x + 1)^2 / (x - 2) * (x + 2)(x - 2) / (x + 1)(x - 1)
Simplify:
(x + 1)(x + 2) / (x - 1)
Therefore, the answer is A. Start Fraction left-parenthesis lower x plus 1 right-parenthesis left-parenthesis lower x plus 2 right-parenthesis over lower x minus 1 End Fraction.
To divide the given expression, follow these steps:
Step 1: Factorize the numerator and the denominator of the fraction.
The numerator: x^2 + 2x + 1 can be factored as (x + 1)(x + 1) = (x + 1)^2.
The denominator: x - 2 can't be further factored.
The numerator of the second fraction: x^2 - 1 can be factored as (x + 1)(x - 1).
The denominator of the second fraction: x^2 - 4 can be factored as (x + 2)(x - 2).
So, the expression becomes [(x + 1)^2/(x - 2)] / [(x + 1)(x - 1)/(x + 2)(x - 2)].
Step 2: Invert the second fraction and multiply it with the first fraction.
[(x + 1)^2/(x - 2)] * [(x + 2)(x - 2)/(x + 1)(x - 1)]
Step 3: Simplify the expression by canceling out common factors.
[(x + 1)/(x - 2)] * [(x + 2)/(x - 1)]
Step 4: Multiply the numerators and the denominators separately.
[(x + 1)(x + 2)] / [(x - 2)(x - 1)]
Step 5: Expand and combine like terms in the numerator and denominator.
(x^2 + 3x + 2) / (x^2 - 3x + 2)
So, the correct answer is:
B. Start Fraction (x^2 + 3x + 2) / (x^2 - 3x + 2) End Fraction