To solve the division of the polynomial [tex]\(\frac{c^3 - 3c^2 + 18c - 16}{c^2 + 3c - 2}\)[/tex], we perform polynomial long division.
### Step-by-Step Long Division
1. Setup the Division:
- Dividend: [tex]\(c^3 - 3c^2 + 18c - 16\)[/tex]
- Divisor: [tex]\(c^2 + 3c - 2\)[/tex]
2. Divide the Leading Terms:
- Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{c^3}{c^2} = c\)[/tex].
- This gives us the first term of the quotient: [tex]\(c\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(c\)[/tex]: [tex]\(c \cdot (c^2 + 3c - 2) = c^3 + 3c^2 - 2c\)[/tex].
- Subtract this product from the original dividend: \\
[tex]\[
(c^3 - 3c^2 + 18c - 16) - (c^3 + 3c^2 - 2c) = -6c^2 + 20c - 16
\][/tex]
4. Repeat the Process with the New Dividend:
- Divide the leading term of the new dividend by the leading term of the divisor: [tex]\(\frac{-6c^2}{c^2} = -6\)[/tex].
- This gives us the next term of the quotient: [tex]\(-6\)[/tex].
5. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(-6\)[/tex]: [tex]\(-6 \cdot (c^2 + 3c - 2) = -6c^2 - 18c + 12\)[/tex].
- Subtract this product from the new dividend: \\
[tex]\[
(-6c^2 + 20c - 16) - (-6c^2 - 18c + 12) = 38c - 28
\][/tex]
6. Write Down the Quotient and Remainder:
- The quotient after the division process is [tex]\(c - 6\)[/tex].
- The remainder is [tex]\(38c - 28\)[/tex].
### Final Expression
Combining these results, we get:
[tex]\[
\frac{c^3 - 3c^2 + 18c - 16}{c^2 + 3c - 2} = (c - 6) + \frac{38c - 28}{c^2 + 3c - 2}
\][/tex]
Thus, the final answer is:
[tex]\[
c - 6 + \frac{38c - 28}{c^2 + 3c - 2}
\][/tex]
In summary, the quotient is [tex]\(c - 6\)[/tex], the remainder is [tex]\(38c - 28\)[/tex], and the complete expression is:
[tex]\[
c + \frac{38c - 28}{c^2 + 3c - 2} - 6
\][/tex]
