Answer :
To find the product of the polynomials [tex]\( (9x^2 - 6x + 1)(3x - 1) \)[/tex], we'll use the distributive property, also known as the FOIL method when dealing with binomials. Let's go through it step-by-step:
1. Distribute Each Term of the First Polynomial to the Second Polynomial:
We'll distribute each term in the first polynomial [tex]\( 9x^2 - 6x + 1 \)[/tex] to each term in the second polynomial [tex]\( 3x - 1 \)[/tex].
2. Multiply [tex]\( 9x^2 \)[/tex] with Each Term in [tex]\( 3x - 1 \)[/tex]:
- [tex]\( 9x^2 \cdot 3x = 27x^3 \)[/tex]
- [tex]\( 9x^2 \cdot (-1) = -9x^2 \)[/tex]
3. Multiply [tex]\( -6x \)[/tex] with Each Term in [tex]\( 3x - 1 \)[/tex]:
- [tex]\(-6x \cdot 3x = -18x^2 \)[/tex]
- [tex]\(-6x \cdot (-1) = 6x \)[/tex]
4. Multiply [tex]\( 1 \)[/tex] with Each Term in [tex]\( 3x - 1 \)[/tex]:
- [tex]\( 1 \cdot 3x = 3x \)[/tex]
- [tex]\( 1 \cdot (-1) = -1 \)[/tex]
5. Combine All the Results:
Now, we add up all these results:
[tex]\[
27x^3 - 9x^2 - 18x^2 + 6x + 3x - 1
\][/tex]
6. Combine Like Terms:
- The [tex]\( x^3 \)[/tex] term is [tex]\( 27x^3 \)[/tex].
- The [tex]\( x^2 \)[/tex] terms are [tex]\(-9x^2 - 18x^2 = -27x^2\)[/tex].
- The [tex]\( x \)[/tex] terms are [tex]\( 6x + 3x = 9x\)[/tex].
- The constant term is [tex]\(-1\)[/tex].
So, when we combine all like terms, we get:
[tex]\[
27x^3 - 27x^2 + 9x - 1
\][/tex]
Thus, the product of the polynomials is [tex]\( 27x^3 - 27x^2 + 9x - 1 \)[/tex], which matches option (C).
1. Distribute Each Term of the First Polynomial to the Second Polynomial:
We'll distribute each term in the first polynomial [tex]\( 9x^2 - 6x + 1 \)[/tex] to each term in the second polynomial [tex]\( 3x - 1 \)[/tex].
2. Multiply [tex]\( 9x^2 \)[/tex] with Each Term in [tex]\( 3x - 1 \)[/tex]:
- [tex]\( 9x^2 \cdot 3x = 27x^3 \)[/tex]
- [tex]\( 9x^2 \cdot (-1) = -9x^2 \)[/tex]
3. Multiply [tex]\( -6x \)[/tex] with Each Term in [tex]\( 3x - 1 \)[/tex]:
- [tex]\(-6x \cdot 3x = -18x^2 \)[/tex]
- [tex]\(-6x \cdot (-1) = 6x \)[/tex]
4. Multiply [tex]\( 1 \)[/tex] with Each Term in [tex]\( 3x - 1 \)[/tex]:
- [tex]\( 1 \cdot 3x = 3x \)[/tex]
- [tex]\( 1 \cdot (-1) = -1 \)[/tex]
5. Combine All the Results:
Now, we add up all these results:
[tex]\[
27x^3 - 9x^2 - 18x^2 + 6x + 3x - 1
\][/tex]
6. Combine Like Terms:
- The [tex]\( x^3 \)[/tex] term is [tex]\( 27x^3 \)[/tex].
- The [tex]\( x^2 \)[/tex] terms are [tex]\(-9x^2 - 18x^2 = -27x^2\)[/tex].
- The [tex]\( x \)[/tex] terms are [tex]\( 6x + 3x = 9x\)[/tex].
- The constant term is [tex]\(-1\)[/tex].
So, when we combine all like terms, we get:
[tex]\[
27x^3 - 27x^2 + 9x - 1
\][/tex]
Thus, the product of the polynomials is [tex]\( 27x^3 - 27x^2 + 9x - 1 \)[/tex], which matches option (C).
