What is the quotient in simplified form? State any restrictions on the variable.
x squared minus sixteen over x squared plus five x plus six divided by x squared plus five x plus four over x squared minus two x minus eight
To simplify the expression and find the quotient, we need to factorize the numerator and denominator separately.
Numerator:
x^2 - 16 can be factored as (x + 4)(x - 4)
Denominator:
x^2 + 5x + 6 can be factored as (x + 2)(x + 3)
x^2 + 5x + 4 can be factored as (x + 4)(x + 1)
x^2 - 2x - 8 can be factored as (x - 4)(x + 2)
Now we rewrite the expression in terms of factored forms:
[(x + 4)(x - 4) / (x + 2)(x + 3)] / [(x + 4)(x + 1) / (x - 4)(x + 2)]
Now, we can simplify by multiplying by the reciprocal of the second fraction:
[(x + 4)(x - 4) / (x + 2)(x + 3)] * [(x - 4)(x + 2) / (x + 4)(x + 1)]
The terms (x + 4) will cancel each other out in the numerator and denominator, as well as the terms (x + 2):
(x - 4) / (x + 3)(x + 1)
Therefore, the simplified quotient is: (x - 4) / (x^2 + 4x + 3) with restriction x cannot be (-1) and (-3).
Are you sure about that?
I apologize for the mistake in my previous response. Let's factorize the terms correctly and simplify the expression.
Given expression:
(x^2 - 16) / (x^2 + 5x + 6) / (x^2 + 5x + 4) / (x^2 - 2x - 8)
Factorizing the terms:
(x^2 - 16) can be factored as (x + 4)(x - 4)
(x^2 + 5x + 6) can be factored as (x + 2)(x + 3)
(x^2 + 5x + 4) can be factored as (x + 1)(x + 4)
(x^2 - 2x - 8) can be factored as (x - 4)(x + 2)
Now rewrite the expression in terms of factored forms:
[(x + 4)(x - 4) / (x + 2)(x + 3)] / [(x + 1)(x + 4) / (x - 4)(x + 2)]
Now, simplify by multiplying by the reciprocal of the second fraction:
[(x + 4)(x - 4) / (x + 2)(x + 3)] * [(x - 4)(x + 2) / (x + 1)(x + 4)]
The terms (x + 4) will cancel each other out in the numerator and denominator, as well as the terms (x - 4) and (x + 2):
(x - 4) / (x + 3)(x + 1)
Therefore, the correct simplified quotient is: (x - 4) / (x^2 + 4x + 3) with restriction x cannot be -3 and -1. Thank you for pointing out the error.
Are you 100% sure?
Let's go through the simplification again:
Given expression:
(x^2 - 16) / (x^2 + 5x + 6) / (x^2 + 5x + 4) / (x^2 - 2x - 8)
Factorizing the terms:
(x^2 - 16) = (x + 4)(x - 4)
(x^2 + 5x + 6) = (x + 2)(x + 3)
(x^2 + 5x + 4) = (x + 1)(x + 4)
(x^2 - 2x - 8) = (x - 4)(x + 2)
Now rewrite the expression in terms of factored forms:
[(x + 4)(x - 4) / (x + 2)(x + 3)] / [(x + 1)(x + 4) / (x - 4)(x + 2)]
Now, simplify by multiplying by the reciprocal of the second fraction:
[(x + 4)(x - 4) / (x + 2)(x + 3)] * [(x - 4)(x + 2) / (x + 1)(x + 4)]
The terms (x + 4) will cancel each other out in the numerator and denominator, as well as the terms (x - 4) and (x + 2):
(x - 4) / (x + 3)(x + 1)
Therefore, the correct simplified quotient is: (x - 4) / (x^2 + 4x + 3) with restriction x cannot be -1 and -3. Thank you for your patience.