High School

Find the product of the polynomials [tex]\left(9x^2 - 6x + 1\right)(3x - 1)[/tex].

A. [tex]27x^3 - 9x^2 + 27x - 1[/tex]
B. [tex]27x^3 - 27x^4 + 9x - 1[/tex]
C. [tex]27x^3 - 27x^2 + 9x - 1[/tex]
D. [tex]27x^3 + 27x^2 + 9x + 1[/tex]

Answer :

To find the product of the polynomials [tex]\((9x^2 - 6x + 1)(3x - 1)\)[/tex], we can use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Let's use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials step-by-step:

### Step-by-Step Calculation:

1. Multiply each term of [tex]\(9x^2 - 6x + 1\)[/tex] by each term of [tex]\(3x - 1\)[/tex]:
- [tex]\(9x^2 \cdot 3x = 27x^3\)[/tex]
- [tex]\(9x^2 \cdot (-1) = -9x^2\)[/tex]
- [tex]\(-6x \cdot 3x = -18x^2\)[/tex]
- [tex]\(-6x \cdot (-1) = 6x\)[/tex]
- [tex]\(1 \cdot 3x = 3x\)[/tex]
- [tex]\(1 \cdot (-1) = -1\)[/tex]

2. Combine all these products:
[tex]\[
27x^3 + (-9x^2) + (-18x^2) + 6x + 3x - 1
\][/tex]

3. Simplify by combining like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 - 18x^2 = -27x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(6x + 3x = 9x\)[/tex]

Putting it all together, we get:
[tex]\[
27x^3 - 27x^2 + 9x - 1
\][/tex]

So, the product of the polynomials [tex]\((9x^2 - 6x + 1)(3x - 1)\)[/tex] is:

[tex]\[
\boxed{27x^3 - 27x^2 + 9x - 1}
\][/tex]

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