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Answer :
Sure! Let's solve the problem of multiplying the polynomials [tex]\((9x + 8)\)[/tex] and [tex]\((3x^2 + x - 1)\)[/tex] step-by-step.
Step 1: Distribute each term in the first polynomial by each term in the second polynomial.
To start, distribute [tex]\(9x\)[/tex] across the second polynomial:
1. [tex]\(9x \cdot 3x^2 = 27x^3\)[/tex]
2. [tex]\(9x \cdot x = 9x^2\)[/tex]
3. [tex]\(9x \cdot (-1) = -9x\)[/tex]
Next, distribute [tex]\(8\)[/tex] across the second polynomial:
1. [tex]\(8 \cdot 3x^2 = 24x^2\)[/tex]
2. [tex]\(8 \cdot x = 8x\)[/tex]
3. [tex]\(8 \cdot (-1) = -8\)[/tex]
Step 2: Combine all the terms.
Now, combine all the results from the distribution:
- [tex]\(27x^3\)[/tex] (from [tex]\(9x \cdot 3x^2\)[/tex])
- [tex]\(9x^2 + 24x^2 = 33x^2\)[/tex] (from [tex]\(9x \cdot x\)[/tex] and [tex]\(8 \cdot 3x^2\)[/tex])
- [tex]\(-9x + 8x = -x\)[/tex] (from [tex]\(9x \cdot (-1)\)[/tex] and [tex]\(8 \cdot x\)[/tex])
- [tex]\(-8\)[/tex] (from [tex]\(8 \cdot (-1)\)[/tex])
Final Result
Thus, combining all these terms, we have the resulting polynomial:
[tex]\[ 27x^3 + 33x^2 - x - 8 \][/tex]
This is the expanded and combined expression from multiplying [tex]\((9x + 8)\)[/tex] by [tex]\((3x^2 + x - 1)\)[/tex].
Step 1: Distribute each term in the first polynomial by each term in the second polynomial.
To start, distribute [tex]\(9x\)[/tex] across the second polynomial:
1. [tex]\(9x \cdot 3x^2 = 27x^3\)[/tex]
2. [tex]\(9x \cdot x = 9x^2\)[/tex]
3. [tex]\(9x \cdot (-1) = -9x\)[/tex]
Next, distribute [tex]\(8\)[/tex] across the second polynomial:
1. [tex]\(8 \cdot 3x^2 = 24x^2\)[/tex]
2. [tex]\(8 \cdot x = 8x\)[/tex]
3. [tex]\(8 \cdot (-1) = -8\)[/tex]
Step 2: Combine all the terms.
Now, combine all the results from the distribution:
- [tex]\(27x^3\)[/tex] (from [tex]\(9x \cdot 3x^2\)[/tex])
- [tex]\(9x^2 + 24x^2 = 33x^2\)[/tex] (from [tex]\(9x \cdot x\)[/tex] and [tex]\(8 \cdot 3x^2\)[/tex])
- [tex]\(-9x + 8x = -x\)[/tex] (from [tex]\(9x \cdot (-1)\)[/tex] and [tex]\(8 \cdot x\)[/tex])
- [tex]\(-8\)[/tex] (from [tex]\(8 \cdot (-1)\)[/tex])
Final Result
Thus, combining all these terms, we have the resulting polynomial:
[tex]\[ 27x^3 + 33x^2 - x - 8 \][/tex]
This is the expanded and combined expression from multiplying [tex]\((9x + 8)\)[/tex] by [tex]\((3x^2 + x - 1)\)[/tex].
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