Question

Factor the expression completely.
minus, 8, x, to the power 5 , plus, x
−8x
5
+x

Answer 1

To factor the expression \(-8x^5 + x\), we start by identifying the common factors in the terms of the expression.

1. **Identify the common factor:**
Both terms \(-8x^5\) and \(x\) have \(x\) as a common factor.

2. **Factor out the common factor:**
We can factor \(x\) out of the expression:

\[
-8x^5 + x = x(-8x^4 + 1)
\]

3. **Recognize if the remaining expression can be factored further:**
The expression inside the parentheses \(-8x^4 + 1\) can be rearranged to \(1 - 8x^4\). This conforms to the difference of squares, as \(1\) can be represented as \(1^2\) and \(8x^4\) can be represented as \((\sqrt{8}x^2)^2\).

4. **Applying the difference of squares:**
Use the formula \(a^2 - b^2 = (a - b)(a + b)\) where \(a = 1\) and \(b = \sqrt{8}x^2\):

\[
1 - 8x^4 = (1 - \sqrt{8}x^2)(1 + \sqrt{8}x^2)
\]

Since \(\sqrt{8} = 2\sqrt{2}\), we can rewrite it as:

\[
1 - 8x^4 = (1 - 2\sqrt{2}x^2)(1 + 2\sqrt{2}x^2)
\]

5. **Combine everything into the final factored form:**
Now, we can write the complete factored form of the original expression:

\[
-8x^5 + x = x(1 - 2\sqrt{2}x^2)(1 + 2\sqrt{2}x^2)
\]

So, the complete factorization of the expression \(-8x^5 + x\) is:

\[
x(1 - 2\sqrt{2}x^2)(1 + 2\sqrt{2}x^2)
\]