Answer :
Let's solve the problem step-by-step using algebraic multiplication (also known as the distributive property):
We need to multiply the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex]:
1. Distribute [tex]\(x^2\)[/tex] to each term in the second polynomial:
[tex]\[
x^2 \cdot 2x^2 = 2x^4
\][/tex]
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
x^2 \cdot (-3) = -3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] to each term in the second polynomial:
[tex]\[
-5x \cdot 2x^2 = -10x^3
\][/tex]
[tex]\[
-5x \cdot x = -5x^2
\][/tex]
[tex]\[
-5x \cdot (-3) = 15x
\][/tex]
3. Combine all the terms:
By adding all the results from steps 1 and 2 together, we get:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
4. Combine like terms:
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
This results in:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
So, the final expression after multiplication is:
[tex]\[ 2x^4 - 9x^3 - 8x^2 + 15x \][/tex]
The correct answer corresponds to option C: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
We need to multiply the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex]:
1. Distribute [tex]\(x^2\)[/tex] to each term in the second polynomial:
[tex]\[
x^2 \cdot 2x^2 = 2x^4
\][/tex]
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
x^2 \cdot (-3) = -3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] to each term in the second polynomial:
[tex]\[
-5x \cdot 2x^2 = -10x^3
\][/tex]
[tex]\[
-5x \cdot x = -5x^2
\][/tex]
[tex]\[
-5x \cdot (-3) = 15x
\][/tex]
3. Combine all the terms:
By adding all the results from steps 1 and 2 together, we get:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
4. Combine like terms:
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
This results in:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
So, the final expression after multiplication is:
[tex]\[ 2x^4 - 9x^3 - 8x^2 + 15x \][/tex]
The correct answer corresponds to option C: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
