Fully simplify the expression below and write your answer as a single fraction.
start fraction, x, cubed, minus, 16, x, divided by, 6, x, plus, 24, end fraction, dot, start fraction, x, squared, minus, 100, divided by, x, to the power 4 , minus, 14, x, cubed, plus, 40, x, squared, end fraction
6x+24
x
3
−16x
⋅
x
4
−14x
3
+40x
2
x
2
−100
(x^3 - 16x) / (6x + 24) * (x^2 - 100) / (x^4 - 14x^3 + 40x^2)
The expression can be simplified as (x^3 - 16x)(x^2 - 100) / (6x + 24)(x^4 - 14x^3 + 40x^2)
Expanding the numerators will give:
(x^5 - 100x^3 - 16x^3 + 160) / (6x + 24)(x^4 - 14x^3 + 40x^2)
(x^5 - 116x^3 + 160) / (6x + 24)(x^4 - 14x^3 + 40x^2)
Factorizing the common term and simplifying:
x(x^4 - 116x^2 + 160) / (6x + 24)(x^4 - 14x^3 + 40x^2)
Further factorizing the term:
x(x^2 - 4)(x^2 - 40) / 6(x + 4)x(x - 10)
Simplifying will give:
(x(x^2 - 4)(x^2 - 40)) / 6(x + 4)x(x - 10)
(x^5 - 4x^3 - 40x^2) / 6(x + 4)(x - 10)
Therefore, the simplified expression is (x^5 - 4x^3 - 40x^2) / (6x + 4)(x - 10) or x^5 - 4x^3 - 40x^2 / 6x^2 + 4x - 60x - 40
x
2
+x
x
3
−x
⋅
x
2
−9x+8
6x
2
+6x−432
Greatest Common Factors:
Greatest Common Factors:
start fraction, x, left bracket, x, squared, minus, 1, right bracket, divided by, x, left bracket, x, plus, 1, right bracket, end fraction, dot, start fraction, 6, left bracket, x, squared, plus, x, minus, 72, right bracket, divided by, x, squared, minus, 9, x, plus, 8, end fraction
x(x+1)
x(x
2
−1)
⋅
x
2
−9x+8
6(x
2
+x−72)
start fraction, x, left bracket, x, minus, 1, right bracket, left bracket, x, plus, 1, right bracket, divided by, x, left bracket, x, plus, 1, right bracket, end fraction, dot, start fraction, 6, left bracket, x, minus, 8, right bracket, left bracket, x, plus, 9, right bracket, divided by, left bracket, x, minus, 1, right bracket, left bracket, x, minus, 8, right bracket, end fraction
x(x+1)
x(x−1)(x+1)
⋅
(x−1)(x−8)
6(x−8)(x+9)
Cancel Exact Binomial Factors:
Cancel Exact Binomial Factors:
Color coded.
start fraction, x, left bracket, x, minus, 1, right bracket, left bracket, x, plus, 1, right bracket, divided by, x, left bracket, x, plus, 1, right bracket, end fraction, dot, start fraction, 6, left bracket, x, minus, 8, right bracket, left bracket, x, plus, 9, right bracket, divided by, left bracket, x, minus, 1, right bracket, left bracket, x, minus, 8, right bracket, end fraction
x(x+1)
x(x−1)(x+1)
⋅
(x−1)(x−8)
6(x−8)(x+9)
start fraction, x, 6, left bracket, x, plus, 9, right bracket, divided by, x, end fraction
x
x6(x+9)
Rearrange:
Rearrange:
start fraction, 6, x, left bracket, x, plus, 9, right bracket, divided by, x, end fraction
x
6x(x+9)
Note: , start fraction, x, divided by, x, end fraction, equals, 1
Note:
x
x
=1
6, left bracket, x, plus, 9, right bracket
6(x+9)
Final Answer
it should have been osolved using this format
To fully simplify the expression and write the answer as a single fraction, we will perform the following steps:
Given expression is: \(\frac{\frac{x(x^3 - x)}{x^2 - 9x + 8}}{\frac{6x^2 + 6x - 432}{x^2 - 9x + 8}}\)
First, factorize the expressions where possible:
Numerator: \(\frac{x(x)(x^2 - 1)}{(x - 1)(x + 1)}\)
Denominator: \(\frac{6(x)(x + 1)(x - 8)}{(x - 8)(x + 9)}\)
Simplify by canceling out exact binomial factors:
\(\frac{x(x - 1)(x + 1)}{(x - 1)(x + 1)}\) • \(\frac{6(x - 8)(x + 9)}{(x - 1)(x - 8)}\)
After canceling the binomial factors that are the same in the numerator and denominator:
\(\frac{x}{1}\) • \(\frac{6(x + 9)}{1}\)
Rearranging the expression gives the final answer:
\(\frac{6x(x + 9)}{x}\)
Therefore, the fully simplified expression is \(\frac{6x(x + 9)}{x}\) or 6(x + 9).
can you clarify your answer
Certainly! Let's clarify the simplified expression:
Given Expression: \(\frac{\frac{x(x^3 - x)}{x^2 - 9x + 8}}{\frac{6x^2 + 6x - 432}{x^2 - 9x + 8}}\)
After simplifying and canceling out the binomial factors:
\(\frac{x}{1} \cdot \frac{6(x + 9)}{1}\)
This simplifies to: \(6(x + 9)\)
Therefore, the fully simplified expression of the initial given expression is \(6(x + 9)\).