Question

Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, minus, 1, right parenthesis, left parenthesis, x, squared, plus, 5, x, minus, 1, right parenthesis
(2x−1)(x
2
+5x−1)

Answer 1

To expand the expression \((2x - 1)(x^2 + 5x - 1)\), we will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first factor with each term in the second factor.

Here’s how to do it step by step:

1. Distribute \(2x\) to each term in the second factor:
\[
2x \cdot (x^2) + 2x \cdot (5x) + 2x \cdot (-1)
\]
This gives:
\[
2x^3 + 10x^2 - 2x
\]

2. Now, distribute \(-1\) to each term in the second factor:
\[
-1 \cdot (x^2) + (-1) \cdot (5x) + (-1) \cdot (-1)
\]
This gives:
\[
-x^2 - 5x + 1
\]

3. Now combine all the terms together:
\[
2x^3 + 10x^2 - 2x - x^2 - 5x + 1
\]

4. Next, combine like terms:
- The \(x^2\) terms: \(10x^2 - x^2 = 9x^2\)
- The \(x\) terms: \(-2x - 5x = -7x\)

So, the final expression is:
\[
2x^3 + 9x^2 - 7x + 1
\]

Thus, the expanded polynomial in standard form is:
\[
\boxed{2x^3 + 9x^2 - 7x + 1}
\]