Q. What is the factored form of \( (2x^{3} + 4x^{2} – x) \)?

Answer

To factor the expression \(2x^{3} + 4x^{2} – x\), look for the greatest common factor (GCF) of all three terms.

Each term contains at least one \(x\):
\(2x^{3} = x(2x^{2})\)
\(4x^{2} = x(4x)\)
\(-x = x(-1)\)

Factoring out the \(x\) gives:
\(x(2x^{2} + 4x – 1)\)

The quadratic \(2x^{2} + 4x – 1\) has discriminant \(b^{2}-4ac = 4^{2}-4\cdot 2\cdot(-1)=24\), which is not a perfect square, so it does not factor further over the rationals.

Final Answer:
\(\boxed{x(2x^{2} + 4x – 1)}\)

Detailed Explanation

Problem: Factor: \(2x^{3}+4x^{2}-x\)

Step 1 – Identify the terms

The expression is a trinomial with three terms:
\[
2x^{3},\qquad 4x^{2},\qquad -x
\]

Step 2 – Find the greatest common factor (GCF)

The coefficients 2, 4, and -1 have no common factor greater than 1, and the variable powers are \(x^{3}, x^{2}, x\). The lowest power of \(x\) present in every term is \(x\). Therefore the GCF is \(x\).

Step 3 – Divide each term by the GCF

\[
2x^{3}\div x = 2x^{2},\qquad 4x^{2}\div x = 4x,\qquad -x\div x = -1
\]

Step 4 – Write the factored form

Factor out \(x\) and place the quotients in parentheses:
\[
2x^{3}+4x^{2}-x = x\bigl(2x^{2}+4x-1\bigr)
\]

Step 5 – Determine if the quadratic factors further

Check the discriminant of \(2x^{2}+4x-1\) where \(a=2,\; b=4,\; c=-1\):
\[
\Delta = b^{2}-4ac = 4^{2}-4(2)(-1)=16+8=24
\]
Since \(\Delta=24\) is not a perfect square, the quadratic does not factor over the integers (or rationals).