Certainly! Let's expand the expression [tex]\((2x + 1)(x^2 - x - 7)\)[/tex] step by step to write it as a polynomial in standard form.
### Step-by-Step Solution:
1. Distribute [tex]\(2x\)[/tex] across the trinomial [tex]\(x^2 - x - 7\)[/tex]:
- [tex]\(2x \cdot x^2 = 2x^3\)[/tex]
- [tex]\(2x \cdot (-x) = -2x^2\)[/tex]
- [tex]\(2x \cdot (-7) = -14x\)[/tex]
So, distributing [tex]\(2x\)[/tex] we get:
[tex]\[
2x^3 - 2x^2 - 14x
\][/tex]
2. Distribute [tex]\(1\)[/tex] across the trinomial [tex]\(x^2 - x - 7\)[/tex]:
- [tex]\(1 \cdot x^2 = x^2\)[/tex]
- [tex]\(1 \cdot (-x) = -x\)[/tex]
- [tex]\(1 \cdot (-7) = -7\)[/tex]
So, distributing [tex]\(1\)[/tex] we get:
[tex]\[
x^2 - x - 7
\][/tex]
3. Combine the results from the two distributions:
[tex]\[
2x^3 - 2x^2 - 14x + x^2 - x - 7
\][/tex]
4. Combine like terms:
- The cubic term is: [tex]\(2x^3\)[/tex]
- The quadratic terms are: [tex]\(-2x^2 + x^2 = -x^2\)[/tex]
- The linear terms are: [tex]\(-14x - x = -15x\)[/tex]
- The constant term is: [tex]\(-7\)[/tex]
Therefore, combining like terms, the expanded expression is:
[tex]\[
2x^3 - x^2 - 15x - 7
\][/tex]
So, the polynomial in standard form is:
[tex]\[
2x^3 - x^2 - 15x - 7
\][/tex]
This is the expanded form of the given expression.
