Answer :
Sure! Let's multiply the given polynomials step by step:
We have:
[tex]\[
(x^2 - 5x)(2x^2 + x - 3)
\][/tex]
To multiply these, we'll use the distributive property (also known as the FOIL method when dealing with binomials, but here we have a binomial multiplied by a trinomial). Each term in the first polynomial should be multiplied by each term in the second polynomial.
Let's start:
1. Multiply [tex]\(x^2\)[/tex] by 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. Multiply [tex]\(-5x\)[/tex] by 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]
Now, let's combine all the results:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
Next, we'll combine like terms:
- [tex]\(2x^4\)[/tex] (no other [tex]\(x^4\)[/tex] terms, so it stays [tex]\(2x^4\)[/tex])
- [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- [tex]\(15x\)[/tex] (no other [tex]\(x\)[/tex] terms, so it stays [tex]\(15x\)[/tex])
Putting it all together, the expression simplifies to:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Therefore, the correct answer is:
D. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
We have:
[tex]\[
(x^2 - 5x)(2x^2 + x - 3)
\][/tex]
To multiply these, we'll use the distributive property (also known as the FOIL method when dealing with binomials, but here we have a binomial multiplied by a trinomial). Each term in the first polynomial should be multiplied by each term in the second polynomial.
Let's start:
1. Multiply [tex]\(x^2\)[/tex] by 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. Multiply [tex]\(-5x\)[/tex] by 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]
Now, let's combine all the results:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
Next, we'll combine like terms:
- [tex]\(2x^4\)[/tex] (no other [tex]\(x^4\)[/tex] terms, so it stays [tex]\(2x^4\)[/tex])
- [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- [tex]\(15x\)[/tex] (no other [tex]\(x\)[/tex] terms, so it stays [tex]\(15x\)[/tex])
Putting it all together, the expression simplifies to:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Therefore, the correct answer is:
D. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
