To divide the polynomial [tex]\(9x^2 - 18x - 7\)[/tex] by [tex]\(3x + 1\)[/tex], we will use polynomial long division. Here are the steps:
### Step 1: Setup the Division
Set up the division similar to how you would with numbers, with [tex]\(9x^2 - 18x - 7\)[/tex] (the dividend) being divided by [tex]\(3x + 1\)[/tex] (the divisor).
### Step 2: Divide the Leading Terms
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{9x^2}{3x} = 3x.
\][/tex]
### Step 3: Multiply and Subtract
Multiply the entire divisor [tex]\(3x + 1\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
3x(3x + 1) = 9x^2 + 3x.
\][/tex]
Subtract this result from the dividend:
[tex]\[
(9x^2 - 18x - 7) - (9x^2 + 3x) = -21x - 7.
\][/tex]
### Step 4: Repeat the Process
Now, repeat the process with [tex]\(-21x - 7\)[/tex]:
Divide the leading term of the new dividend by the leading term of the divisor:
[tex]\[
\frac{-21x}{3x} = -7.
\][/tex]
Multiply the entire divisor [tex]\(3x + 1\)[/tex] by [tex]\(-7\)[/tex]:
[tex]\[
-7(3x + 1) = -21x - 7.
\][/tex]
Subtract this result from the new dividend:
[tex]\[
(-21x - 7) - (-21x - 7) = 0.
\][/tex]
### Step 5: Write the Quotient
Since the remainder is 0, there is no remainder. Thus, the quotient is the result of the division:
[tex]\[
3x - 7.
\][/tex]
### Final Answer
The quotient of [tex]\(\frac{9x^2 - 18x - 7}{3x + 1}\)[/tex] is:
[tex]\[
\boxed{3x - 7}.
\][/tex]
