Perform the following operation and express in simplest form.
start fraction, x, cubed, divided by, x, squared, minus, 4, end fraction, divided by, start fraction, 2, x, divided by, x, squared, minus, 7, x, plus, 10, end fraction
x
2
−4
x
3
÷
x
2
−7x+10
2x
To simplify this expression, we need to combine the fractions and simplify the resulting fraction.
Start by simplifying the numerator (x^3).
Next, we can simplify the denominator (x^2 - 4) by factoring it as (x + 2)(x - 2).
Start fraction, x^3, divided by, (x + 2)(x - 2), divided by, (2x)/(x^2 - 7x + 10).
Now we can cancel out common factors between the numerator of the first fraction and the denominator of the second fraction, as well as the denominator of the first fraction and the numerator of the second fraction.
(x^3)/(x + 2)(x - 2) ÷ (2x)/(x^2 - 7x + 10) = (x^3)/(2x(x^2 - 7x + 10))
Next, we can simplify the resulting fraction (x^3)/(2x(x^2 - 7x + 10)) by canceling out the common factor of x in the numerator and denominator.
(x^3)/(2x(x^2 - 7x + 10)) = (x^2)/(2(x^2 - 7x + 10))
Therefore, the simplified expression is (x^2)/(2(x^2 - 7x + 10)).
To simplify the given expression, let's break it down step by step.
Step 1: Simplify the numerator
The numerator is expressed as x^3 divided by (x^2 - 4). To simplify further, we factorize the denominator:
x^3 / (x^2 - 4) = x^3 / (x - 2)(x + 2)
Step 2: Simplify the denominator
The denominator is expressed as (2x) / (x^2 - 7x + 10). To simplify further, we factorize the denominator:
(2x) / (x^2 - 7x + 10) = (2x) / (x - 5)(x - 2)
Step 3: Combine the fractions
Now that we have simplified the numerator and denominator, we can combine the fractions:
(x^3 / (x - 2)(x + 2)) / ((2x) / (x - 5)(x - 2))
Since we have a fraction divided by another fraction, we can simplify it by multiplying the first fraction by the reciprocal of the second fraction:
(x^3 / (x - 2)(x + 2)) * ((x - 5)(x - 2) / (2x))
Step 4: Simplify further
Now, let's simplify the expression further by canceling out common factors. We can cancel the (x - 2) terms from the numerator and denominator:
x^3 * (x - 5) / (2x)
Step 5: Express in simplest form
Finally, we can multiply the remaining terms in the numerator:
(x^4 - 5x^3) / (2x)
Thus, the given expression simplified to its simplest form is (x^4 - 5x^3) / (2x).
To solve the given expression, follow these steps:
Step 1: Simplify the numerator:
The numerator is x cubed divided by x squared minus 4.
To simplify this, divide each term in the numerator by x squared minus 4:
(x cubed) / (x squared minus 4) = (x * x * x) / (x * x - 2 * 2)
= (x * x * x) / ((x + 2) * (x - 2))
Step 2: Simplify the denominator:
The denominator is 2x divided by x squared minus 7x plus 10.
To simplify this, divide each term in the denominator by 2x:
(2x) / (x squared minus 7x plus 10) = (2x) / (x * x - 5x - 2x + 10)
= (2x) / [(x * (x - 5)) - 2 * (x - 5)]
= (2x) / [(x - 5) * (x - 2)]
Step 3: Combine the numerator and denominator:
Now that we have simplified the numerator and denominator separately, we can express the given expression in simplest form by combining them:
[(x * x * x) / ((x + 2) * (x - 2))] ÷ [(2x) / [(x - 5) * (x - 2)]]
Step 4: Invert and multiply:
To divide fractions, we need to invert the second fraction (denominator) and then multiply the two fractions together:
[(x * x * x) / ((x + 2) * (x - 2))] * [ [(x - 5) * (x - 2)] / (2x) ]
= [(x * x * x) * [(x - 5) * (x - 2)]] / [((x + 2) * (x - 2)) * (2x)]
Step 5: Simplify the expression:
Multiplying the numerators and denominators, we get:
= (x^3 * (x - 5) * (x - 2)) / ((x + 2) * (x - 2) * 2x)
Step 6: Further simplify the expression:
Cancel out the common factors between the numerator and denominator:
= (x^2 * (x - 5)) / ((x + 2) * 2)
Final Answer: The expression (x^3 / (x^2 - 4)) / (2x / (x^2 - 7x + 10)) simplifies to:
(x^2 * (x - 5)) / ((x + 2) * 2)