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Answer :
To multiply the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex], we'll apply the distributive property, also known as the FOIL method for binomials, even though one of the polynomials has three terms.
Here are the steps:
1. Distribute [tex]\(x^2\)[/tex] to each term in the second polynomial [tex]\((2x^2 + x - 3)\)[/tex]:
[tex]\[
x^2 \times 2x^2 = 2x^4
\][/tex]
[tex]\[
x^2 \times x = x^3
\][/tex]
[tex]\[
x^2 \times (-3) = -3x^2
\][/tex]
So, the expression becomes:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] to each term in the second polynomial [tex]\((2x^2 + x - 3)\)[/tex]:
[tex]\[
-5x \times 2x^2 = -10x^3
\][/tex]
[tex]\[
-5x \times x = -5x^2
\][/tex]
[tex]\[
-5x \times (-3) = 15x
\][/tex]
So, the expression becomes:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Combine all the results:
Put together all the terms from steps 1 and 2:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
Putting it all together, we get:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Therefore, the correct answer is D. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
Here are the steps:
1. Distribute [tex]\(x^2\)[/tex] to each term in the second polynomial [tex]\((2x^2 + x - 3)\)[/tex]:
[tex]\[
x^2 \times 2x^2 = 2x^4
\][/tex]
[tex]\[
x^2 \times x = x^3
\][/tex]
[tex]\[
x^2 \times (-3) = -3x^2
\][/tex]
So, the expression becomes:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] to each term in the second polynomial [tex]\((2x^2 + x - 3)\)[/tex]:
[tex]\[
-5x \times 2x^2 = -10x^3
\][/tex]
[tex]\[
-5x \times x = -5x^2
\][/tex]
[tex]\[
-5x \times (-3) = 15x
\][/tex]
So, the expression becomes:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Combine all the results:
Put together all the terms from steps 1 and 2:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
Putting it all together, we get:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Therefore, the correct answer is D. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
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