Question
express in simplest form (3x^2-3x-60/x6^2 -2x-24) /( 3x^2 -15x/6x-x^2)
Answers
GPT-4o mini
To simplify the expression
\[
\frac{\frac{3x^2 - 3x - 60}{x^6 - 2x - 24}}{\frac{3x^2 - 15x}{6x - x^2}},
\]
we first rewrite it as:
\[
\frac{3x^2 - 3x - 60}{x^6 - 2x - 24} \cdot \frac{6x - x^2}{3x^2 - 15x}.
\]
Next, we will factor each component of the expression.
**Step 1: Factor \(3x^2 - 3x - 60\)**
Factor out the common term:
\[
3(x^2 - x - 20).
\]
Now, factor \(x^2 - x - 20\):
\[
x^2 - x - 20 = (x - 5)(x + 4).
\]
So,
\[
3x^2 - 3x - 60 = 3(x - 5)(x + 4).
\]
**Step 2: Factor \(x^6 - 2x - 24\)**
This expression requires testing for roots or using polynomial division, but for simplicity, let's assume it factors. The actual factorization can be determined by trial, synthetic division or numeric methods. We'll consider this step temporarily pending additional data but keep our focus on \(6x - x^2\) and \(3x^2 - 15x\).
**Step 3: Factor \(6x - x^2\)**
Factor out the common term:
\[
-x^2 + 6x = -x(x - 6).
\]
**Step 4: Factor \(3x^2 - 15x\)**
Factor out the common term:
\[
3x(x - 5).
\]
**Step 5: Substitute back into the expression**
Substituting back all the factors we have, the expression now looks like:
\[
\frac{3(x - 5)(x + 4)}{x^6 - 2x - 24} \cdot \frac{-x(x - 6)}{3x(x - 5)}.
\]
**Step 6: Cancel out common factors**
In the expression:
1. The \(3x\) in the numerator \(3\) and \(3x\) in the denominator \(3x(x - 5)\) cancels,
2. The \((x - 5)\) cancels as well.
Thus, simplifying this yields:
\[
\frac{-(x + 4)(x - 6)}{x^6 - 2x - 24}.
\]
**Note: Completing the factorization of \(x^6 - 2x - 24\) is complex and not straightforward without specific numerical approaches or additional context.**
Assuming you would verify or evaluate \(x^6 - 2x - 24\), the simplified expression above is as follows, given further detail. Thus, we have:
\[
\boxed{\frac{-(x + 4)(x - 6)}{x^6 - 2x - 24}}.
\]
Otherwise, please evaluate \(x^6 - 2x - 24\) separately (root finding or polynomial division) if simplifying further is necessary!
\[
\frac{\frac{3x^2 - 3x - 60}{x^6 - 2x - 24}}{\frac{3x^2 - 15x}{6x - x^2}},
\]
we first rewrite it as:
\[
\frac{3x^2 - 3x - 60}{x^6 - 2x - 24} \cdot \frac{6x - x^2}{3x^2 - 15x}.
\]
Next, we will factor each component of the expression.
**Step 1: Factor \(3x^2 - 3x - 60\)**
Factor out the common term:
\[
3(x^2 - x - 20).
\]
Now, factor \(x^2 - x - 20\):
\[
x^2 - x - 20 = (x - 5)(x + 4).
\]
So,
\[
3x^2 - 3x - 60 = 3(x - 5)(x + 4).
\]
**Step 2: Factor \(x^6 - 2x - 24\)**
This expression requires testing for roots or using polynomial division, but for simplicity, let's assume it factors. The actual factorization can be determined by trial, synthetic division or numeric methods. We'll consider this step temporarily pending additional data but keep our focus on \(6x - x^2\) and \(3x^2 - 15x\).
**Step 3: Factor \(6x - x^2\)**
Factor out the common term:
\[
-x^2 + 6x = -x(x - 6).
\]
**Step 4: Factor \(3x^2 - 15x\)**
Factor out the common term:
\[
3x(x - 5).
\]
**Step 5: Substitute back into the expression**
Substituting back all the factors we have, the expression now looks like:
\[
\frac{3(x - 5)(x + 4)}{x^6 - 2x - 24} \cdot \frac{-x(x - 6)}{3x(x - 5)}.
\]
**Step 6: Cancel out common factors**
In the expression:
1. The \(3x\) in the numerator \(3\) and \(3x\) in the denominator \(3x(x - 5)\) cancels,
2. The \((x - 5)\) cancels as well.
Thus, simplifying this yields:
\[
\frac{-(x + 4)(x - 6)}{x^6 - 2x - 24}.
\]
**Note: Completing the factorization of \(x^6 - 2x - 24\) is complex and not straightforward without specific numerical approaches or additional context.**
Assuming you would verify or evaluate \(x^6 - 2x - 24\), the simplified expression above is as follows, given further detail. Thus, we have:
\[
\boxed{\frac{-(x + 4)(x - 6)}{x^6 - 2x - 24}}.
\]
Otherwise, please evaluate \(x^6 - 2x - 24\) separately (root finding or polynomial division) if simplifying further is necessary!