To find the product of the expression [tex]\((3x - a)(3x^2 - 4x + 7)\)[/tex], we will use the distributive property. This involves distributing each term in the first polynomial to every term in the second polynomial and then combining like terms.
Given the expressions:
[tex]\[ (3x - a) \][/tex]
[tex]\[ (3x^2 - 4x + 7) \][/tex]
We will distribute each term in the first polynomial to every term in the second polynomial.
### Step 1: Distribute [tex]\(3x\)[/tex]
Distribute [tex]\(3x\)[/tex] to each term in the second polynomial [tex]\((3x^2 - 4x + 7)\)[/tex]:
[tex]\[
3x \cdot 3x^2 = 9x^3
\][/tex]
[tex]\[
3x \cdot (-4x) = -12x^2
\][/tex]
[tex]\[
3x \cdot 7 = 21x
\][/tex]
This gives us:
[tex]\[ 9x^3 - 12x^2 + 21x \][/tex]
### Step 2: Distribute [tex]\(-a\)[/tex]
Distribute [tex]\(-a\)[/tex] to each term in the second polynomial:
[tex]\[
-a \cdot 3x^2 = -3ax^2
\][/tex]
[tex]\[
-a \cdot (-4x) = 4ax
\][/tex]
[tex]\[
-a \cdot 7 = -7a
\][/tex]
This gives us:
[tex]\[-3ax^2 + 4ax - 7a\][/tex]
### Step 3: Combine all the terms
Now, combine all the distributed terms together:
[tex]\[
9x^3 - 12x^2 + 21x - 3ax^2 + 4ax - 7a
\][/tex]
### Step 4: Combine like terms
Group like terms, but in this case, they are already grouped:
[tex]\[
9x^3 + (-12x^2 - 3ax^2) + (21x + 4ax) - 7a
\][/tex]
Which results in the expression:
[tex]\[
9x^3 - 3ax^2 - 12x^2 + 4ax + 21x - 7a
\][/tex]
Thus, the product of [tex]\((3x - a)(3x^2 - 4x + 7)\)[/tex] is:
[tex]\[
9x^3 - 3ax^2 - 12x^2 + 4ax + 21x - 7a
\][/tex]
So the final simplified product is:
[tex]\[
\boxed{9x^3 - 3ax^2 - 12x^2 + 4ax + 21x - 7a}
\][/tex]
