Answer :
The factored form of the expression is: \(2x^5(x-1)^2(x+3)\). You can check the factoring by multiplying it out.
To factor the expression \(2x^8 + 2x^7 - 14x^6 + 4x^5\), we can first look for common factors among the terms.
Taking out the common factor of \(2x^5\), we get:
\(2x^5(x^3 + x^2 - 7x + 2)\)
Now, we can focus on factoring the quadratic expression \(x^3 + x^2 - 7x + 2\). To do this, we can look for possible rational roots using the Rational Root Theorem or try factoring by grouping.
If we inspect the quadratic expression further, we can notice that \(x = 1\) is a root. By performing synthetic division, we find:
\[
\begin{array}{c|cccc}
1 & 1 & 1 & -7 & 2 \\
\hline
& & 1 & 2 & -5 \\
\end{array}
\]
The result tells us that \(x - 1\) is a factor. Dividing the original expression by \(x - 1\), we obtain:
\(2x^5(x - 1)(x^2 + 2x - 5)\)
To factor \(x^2 + 2x - 5\), we can either use the quadratic formula or look for two numbers whose sum is 2 and whose product is -5. By factoring, we find:
\(x^2 + 2x - 5 = (x - 1)(x + 3)\)
Therefore, the factored form of the expression \(2x^8 + 2x^7 - 14x^6 + 4x^5\) is:
\(2x^5(x - 1)(x - 1)(x + 3)\)
To check the factoring, we can multiply the factors back together:
\(2x^5(x - 1)(x - 1)(x + 3) = 2x^5(x^2 - 2x + 1)(x + 3)\)
Expanding further:
\(2x^5(x^3 - 2x^2 + x + 3x^2 - 6x + 3) = 2x^5(x^3 + x^2 - 5x + 3)\)
This matches the original expression, so the factoring is correct.
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