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Answer :
To solve the problem of multiplying and simplifying [tex]\((3x + 5)(2x^2 - 3x + 2)\)[/tex], we can follow these steps:
1. Distribute Each Term in the First Polynomial:
- Multiply [tex]\(3x\)[/tex] by each term in the trinomial [tex]\(2x^2 - 3x + 2\)[/tex].
- Multiply [tex]\(5\)[/tex] by each term in the trinomial [tex]\(2x^2 - 3x + 2\)[/tex].
2. Perform the Multiplications:
- [tex]\(3x \times 2x^2 = 6x^3\)[/tex]
- [tex]\(3x \times -3x = -9x^2\)[/tex]
- [tex]\(3x \times 2 = 6x\)[/tex]
- [tex]\(5 \times 2x^2 = 10x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
3. Combine All the Results Together:
- Combining the results from the multiplications, we have:
[tex]\[
6x^3 - 9x^2 + 6x + 10x^2 - 15x + 10
\][/tex]
4. Simplify by Combining Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 + 10x^2 = x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(6x - 15x = -9x\)[/tex]
5. Write the Final, Simplified Expression:
[tex]\[
6x^3 + x^2 - 9x + 10
\][/tex]
So, the simplified result of multiplying [tex]\((3x + 5)(2x^2 - 3x + 2)\)[/tex] is [tex]\(6x^3 + x^2 - 9x + 10\)[/tex].
1. Distribute Each Term in the First Polynomial:
- Multiply [tex]\(3x\)[/tex] by each term in the trinomial [tex]\(2x^2 - 3x + 2\)[/tex].
- Multiply [tex]\(5\)[/tex] by each term in the trinomial [tex]\(2x^2 - 3x + 2\)[/tex].
2. Perform the Multiplications:
- [tex]\(3x \times 2x^2 = 6x^3\)[/tex]
- [tex]\(3x \times -3x = -9x^2\)[/tex]
- [tex]\(3x \times 2 = 6x\)[/tex]
- [tex]\(5 \times 2x^2 = 10x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
3. Combine All the Results Together:
- Combining the results from the multiplications, we have:
[tex]\[
6x^3 - 9x^2 + 6x + 10x^2 - 15x + 10
\][/tex]
4. Simplify by Combining Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 + 10x^2 = x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(6x - 15x = -9x\)[/tex]
5. Write the Final, Simplified Expression:
[tex]\[
6x^3 + x^2 - 9x + 10
\][/tex]
So, the simplified result of multiplying [tex]\((3x + 5)(2x^2 - 3x + 2)\)[/tex] is [tex]\(6x^3 + x^2 - 9x + 10\)[/tex].
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