Perform the following operation and express in simplest form.
start fraction, 2, x, minus, 10, divided by, x, plus, 8, end fraction, dot, start fraction, x, squared, minus, 64, divided by, x, squared, plus, 3, x, minus, 40, end fraction
x+8
2x−10
⋅
x
2
+3x−40
x
2
−64
To simplify the expression, we first find the LCD of the two fractions in the numerator and denominator.
The factors of the denominators are:
x + 8 = (x + 4)(x + 2)
x^2 + 3x - 40 = (x + 8)(x - 5)
So, the LCD is (x + 4)(x + 2)(x + 8)(x - 5).
Now, we can rewrite the expression with the common denominator:
(2x - 10)(x + 4)(x + 2)
_______________________
x(x + 8)(x - 5)(x^2 - 64)
Next, we can factor the numerator and denominator further:
2(x - 5)(x + 4)(x + 2)
_______________________
x(x + 8)(x - 5)(x + 8)(x - 8)
From here, we can cancel out the common factors:
2(x + 4)(x + 2)
___________________
x(x + 8)(x - 8)
Thus, the simplified expression is: 2(x + 4)(x + 2) / (x)(x + 8)(x - 8)
To perform the given operation and simplify the expression, we need to multiply the two fractions:
(2x - 10)/(x + 8) * (x^2 - 64)/(x^2 + 3x - 40)
First, let's factorize the numerator and denominator:
Numerator:
x^2 - 64 = (x - 8)(x + 8)
Denominator:
x^2 + 3x - 40 = (x + 8)(x - 5)
Now, we can cancel out the common factors:
((2x - 10) / (x + 8)) * ((x - 8)(x + 8) / (x + 8)(x - 5))
Simplifying further:
(2x - 10) * (x - 8) / (x - 5)
Expanding the numerator:
(2x * x - 16x - 10x + 80) / (x - 5)
Combining like terms:
(2x^2 - 26x + 80) / (x - 5)
Therefore, the expression simplifies to:
(2x^2 - 26x + 80) / (x - 5)
To perform the operation and express it in the simplest form, let's follow these steps:
1. Start by factoring the numerator and denominator:
- Numerator: 2x - 10 = 2(x - 5)
- Denominator: x^2 + 3x - 40 = (x - 5)(x + 8)
2. Simplify the expression:
The fraction now becomes:
(2(x - 5) / (x - 5)(x + 8) ) * (x^2 - 64 / x^2 + 3x - 40)
3. Cancel out like terms:
We can cancel out the (x - 5) terms in the numerator and denominator:
(2 / (x + 8) ) * (x^2 - 64 / x^2 + 3x - 40)
4. Factor the numerator:
The numerator can be factored as the difference of squares:
x^2 - 64 = (x - 8)(x + 8)
5. Rewrite the expression:
The expression becomes:
(2 / (x + 8) ) * ((x - 8)(x + 8) / x^2 + 3x - 40)
6. Cancel out common factors:
We can cancel out the (x + 8) terms in the numerator and denominator:
The simplified expression is:
2(x - 8) / (x^2 + 3x - 40)
Therefore, the final expression in simplest form is 2(x - 8) / (x^2 + 3x - 40).