Answer :
Sure, let's find the product of the polynomials [tex]\((9x^2 - 6x + 1)(3x - 1)\)[/tex] step-by-step.
We need to use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
The given polynomials are:
[tex]\[ (9x^2 - 6x + 1)(3x - 1) \][/tex]
Step-by-step multiplication:
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]
4. Combine all the terms:
[tex]\[
27x^3 - 9x^2 - 18x^2 + 6x + 3x - 1
\][/tex]
5. Simplify by combining like terms:
[tex]\[
27x^3 - (9x^2 + 18x^2) + (6x + 3x) - 1
\][/tex]
[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]
Among the given options, this matches with:
[tex]\[ 27x^3 - 27x^2 + 9x - 1\][/tex]
Therefore, the correct answer is:
[tex]\[\boxed{27x^3 - 27x^2 + 9x - 1}\][/tex]
We need to use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
The given polynomials are:
[tex]\[ (9x^2 - 6x + 1)(3x - 1) \][/tex]
Step-by-step multiplication:
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]
4. Combine all the terms:
[tex]\[
27x^3 - 9x^2 - 18x^2 + 6x + 3x - 1
\][/tex]
5. Simplify by combining like terms:
[tex]\[
27x^3 - (9x^2 + 18x^2) + (6x + 3x) - 1
\][/tex]
[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]
Among the given options, this matches with:
[tex]\[ 27x^3 - 27x^2 + 9x - 1\][/tex]
Therefore, the correct answer is:
[tex]\[\boxed{27x^3 - 27x^2 + 9x - 1}\][/tex]
