To expand the expression [tex]\((3x + 10)(-3x^2 + x - 3)\)[/tex] into a polynomial in standard form, follow these steps:
1. Distribute each term in the first polynomial to every term in the second polynomial:
[tex]\[
(3x + 10)(-3x^2 + x - 3)
\][/tex]
This means we will distribute [tex]\(3x\)[/tex] to each term in [tex]\(-3x^2 + x - 3\)[/tex] and then distribute [tex]\(10\)[/tex] to each term in [tex]\(-3x^2 + x - 3\)[/tex].
2. Distribute [tex]\(3x\)[/tex] to [tex]\(-3x^2\)[/tex]:
[tex]\[
3x \cdot (-3x^2) = -9x^3
\][/tex]
3. Distribute [tex]\(3x\)[/tex] to [tex]\(x\)[/tex]:
[tex]\[
3x \cdot x = 3x^2
\][/tex]
4. Distribute [tex]\(3x\)[/tex] to [tex]\(-3\)[/tex]:
[tex]\[
3x \cdot (-3) = -9x
\][/tex]
5. Distribute [tex]\(10\)[/tex] to [tex]\(-3x^2\)[/tex]:
[tex]\[
10 \cdot (-3x^2) = -30x^2
\][/tex]
6. Distribute [tex]\(10\)[/tex] to [tex]\(x\)[/tex]:
[tex]\[
10 \cdot x = 10x
\][/tex]
7. Distribute [tex]\(10\)[/tex] to [tex]\(-3\)[/tex]:
[tex]\[
10 \cdot (-3) = -30
\][/tex]
8. Combine all the terms from the distributions:
[tex]\[
-9x^3 + 3x^2 - 9x - 30x^2 + 10x - 30
\][/tex]
9. Combine like terms:
- First, combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
3x^2 - 30x^2 = -27x^2
\][/tex]
- Next, combine the [tex]\(x\)[/tex] terms:
[tex]\[
-9x + 10x = x
\][/tex]
10. Write the final expanded polynomial in standard form:
[tex]\[
-9x^3 - 27x^2 + x - 30
\][/tex]
Therefore, the expanded polynomial in standard form is:
[tex]\[
-9x^3 - 27x^2 + x - 30
\][/tex]
