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Answer :
Sure! Let's find the product of the polynomials [tex]\((9x^2 - 6x + 1)(3x - 1)\)[/tex] step-by-step.
### Step 1: Distribute the Terms
We'll use the distributive property (also known as the distributive law of multiplication) to multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\(9x^2\)[/tex] by each term in [tex]\(3x - 1\)[/tex]:
- [tex]\(9x^2 \cdot 3x = 27x^3\)[/tex]
- [tex]\(9x^2 \cdot (-1) = -9x^2\)[/tex]
2. Multiply [tex]\(-6x\)[/tex] by each term in [tex]\(3x - 1\)[/tex]:
- [tex]\(-6x \cdot 3x = -18x^2\)[/tex]
- [tex]\(-6x \cdot (-1) = 6x\)[/tex]
3. Multiply [tex]\(1\)[/tex] by each term in [tex]\(3x - 1\)[/tex]:
- [tex]\(1 \cdot 3x = 3x\)[/tex]
- [tex]\(1 \cdot (-1) = -1\)[/tex]
### Step 2: Combine Like Terms
Now, we add up all the terms from the multiplication:
- The [tex]\(x^3\)[/tex] term is [tex]\(27x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 + (-18x^2) = -27x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(6x + 3x = 9x\)[/tex].
- The constant term is [tex]\(-1\)[/tex].
So, the product of the polynomials is:
[tex]\[27x^3 - 27x^2 + 9x - 1\][/tex]
This matches the option [tex]\(\boxed{27x^3 - 27x^2 + 9x - 1}\)[/tex] from the choices provided.
### Step 1: Distribute the Terms
We'll use the distributive property (also known as the distributive law of multiplication) to multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\(9x^2\)[/tex] by each term in [tex]\(3x - 1\)[/tex]:
- [tex]\(9x^2 \cdot 3x = 27x^3\)[/tex]
- [tex]\(9x^2 \cdot (-1) = -9x^2\)[/tex]
2. Multiply [tex]\(-6x\)[/tex] by each term in [tex]\(3x - 1\)[/tex]:
- [tex]\(-6x \cdot 3x = -18x^2\)[/tex]
- [tex]\(-6x \cdot (-1) = 6x\)[/tex]
3. Multiply [tex]\(1\)[/tex] by each term in [tex]\(3x - 1\)[/tex]:
- [tex]\(1 \cdot 3x = 3x\)[/tex]
- [tex]\(1 \cdot (-1) = -1\)[/tex]
### Step 2: Combine Like Terms
Now, we add up all the terms from the multiplication:
- The [tex]\(x^3\)[/tex] term is [tex]\(27x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 + (-18x^2) = -27x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(6x + 3x = 9x\)[/tex].
- The constant term is [tex]\(-1\)[/tex].
So, the product of the polynomials is:
[tex]\[27x^3 - 27x^2 + 9x - 1\][/tex]
This matches the option [tex]\(\boxed{27x^3 - 27x^2 + 9x - 1}\)[/tex] from the choices provided.
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