Answer :
Let's work through the problem step-by-step to find the result of multiplying [tex]\(-3x\)[/tex] by [tex]\(x^3 + 3x^2 - x - 5\)[/tex].
1. Distribute [tex]\(-3x\)[/tex] across each term in the polynomial [tex]\(x^3 + 3x^2 - x - 5\)[/tex]:
- Multiply [tex]\(-3x\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
-3x \cdot x^3 = -3x^4
\][/tex]
- Multiply [tex]\(-3x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
-3x \cdot 3x^2 = -9x^3
\][/tex]
- Multiply [tex]\(-3x\)[/tex] by [tex]\(-x\)[/tex]:
[tex]\[
-3x \cdot (-x) = 3x^2
\][/tex]
- Multiply [tex]\(-3x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
-3x \cdot (-5) = 15x
\][/tex]
2. Combine all the results together:
The expanded polynomial is:
[tex]\[
-3x^4 - 9x^3 + 3x^2 + 15x
\][/tex]
3. Match the expanded expression with the given options:
The correct expression matches option:
- a. [tex]\(-3x^4 - 9x^3 + 3x^2 + 15x\)[/tex]
Therefore, the answer is a. [tex]\(-3x^4 - 9x^3 + 3x^2 + 15x\)[/tex].
1. Distribute [tex]\(-3x\)[/tex] across each term in the polynomial [tex]\(x^3 + 3x^2 - x - 5\)[/tex]:
- Multiply [tex]\(-3x\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
-3x \cdot x^3 = -3x^4
\][/tex]
- Multiply [tex]\(-3x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
-3x \cdot 3x^2 = -9x^3
\][/tex]
- Multiply [tex]\(-3x\)[/tex] by [tex]\(-x\)[/tex]:
[tex]\[
-3x \cdot (-x) = 3x^2
\][/tex]
- Multiply [tex]\(-3x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
-3x \cdot (-5) = 15x
\][/tex]
2. Combine all the results together:
The expanded polynomial is:
[tex]\[
-3x^4 - 9x^3 + 3x^2 + 15x
\][/tex]
3. Match the expanded expression with the given options:
The correct expression matches option:
- a. [tex]\(-3x^4 - 9x^3 + 3x^2 + 15x\)[/tex]
Therefore, the answer is a. [tex]\(-3x^4 - 9x^3 + 3x^2 + 15x\)[/tex].
