To factor the expression \(-8x^5 + x\), we start by identifying the common factors in the terms of the expression.
-
Identify the common factor: Both terms \(-8x^5\) and \(x\) have \(x\) as a common factor.
-
Factor out the common factor: We can factor \(x\) out of the expression:
\[ -8x^5 + x = x(-8x^4 + 1) \]
-
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 \((\sqrt{8}x^2)^2\).
-
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:
\[ 1 - 8x^4 = (1 - 2\sqrt{2}x^2)(1 + 2\sqrt{2}x^2) \]
-
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:
\[ x(1 - 2\sqrt{2}x^2)(1 + 2\sqrt{2}x^2) \]
1 answer