To expand the expression [tex]\((2x - 7)(2x^2 + 2x - 1)\)[/tex] into a polynomial in standard form, we will use the distributive property (also known as the FOIL method for binomials in this context). Let's break it down step-by-step:
First, we start by distributing [tex]\(2x\)[/tex] to each term inside the second parenthesis:
[tex]\[
2x \cdot (2x^2 + 2x - 1)
\][/tex]
This will result in:
[tex]\[
2x \cdot 2x^2 + 2x \cdot 2x + 2x \cdot (-1)
\][/tex]
Simplifying each term, we get:
[tex]\[
4x^3 + 4x^2 - 2x
\][/tex]
Next, we distribute [tex]\(-7\)[/tex] to each term inside the second parenthesis:
[tex]\[
-7 \cdot (2x^2 + 2x - 1)
\][/tex]
This will give us:
[tex]\[
-7 \cdot 2x^2 + (-7) \cdot 2x + (-7) \cdot (-1)
\][/tex]
Simplifying each term, we get:
[tex]\[
-14x^2 - 14x + 7
\][/tex]
Now, we combine the results from both distributions:
[tex]\[
4x^3 + 4x^2 - 2x - 14x^2 - 14x + 7
\][/tex]
Next, we combine like terms. We have:
- [tex]\(4x^2\)[/tex] and [tex]\(-14x^2\)[/tex]
- [tex]\(-2x\)[/tex] and [tex]\(-14x\)[/tex]
- The constant term [tex]\(7\)[/tex] remains as it is.
Combining these like terms, we get:
[tex]\[
4x^3 - 10x^2 - 16x + 7
\][/tex]
Thus, the expanded expression in standard form is:
[tex]\[
4x^3 - 10x^2 - 16x + 7
\][/tex]
