To use synthetic division to divide the polynomial [tex]\(3x^3 + 4x^2 - 7x + 6\)[/tex] by [tex]\(x - 3\)[/tex], we can follow these steps:
1. Write down the coefficients of the polynomial: [tex]\(3, 4, -7, 6\)[/tex].
2. Set up the synthetic division by writing the root corresponding to [tex]\(x - 3\)[/tex], which is [tex]\(3\)[/tex], on the left side of the division.
3. Begin the synthetic division process:
[tex]\[
\begin{array}{r|rrrr}
3 & 3 & 4 & -7 & 6 \\
\hline
& 3 & 13 & 32 & 102 \\
\end{array}
\][/tex]
- Start with the first coefficient, which is [tex]\(3\)[/tex]. Write this number below the division line.
- Multiply this number by the root (which is [tex]\(3\)[/tex]) and add to the next coefficient:
- [tex]\(3 \times 3 = 9\)[/tex]
- [tex]\(4 + 9 = 13\)[/tex]
- Write [tex]\(13\)[/tex] below the division line.
- Repeat the process:
- [tex]\(13 \times 3 = 39\)[/tex]
- [tex]\(-7 + 39 = 32\)[/tex]
- Write [tex]\(32\)[/tex] below the division line.
- Repeat the process again:
- [tex]\(32 \times 3 = 96\)[/tex]
- [tex]\(6 + 96 = 102\)[/tex]
- Write [tex]\(102\)[/tex] below the division line.
4. The last number [tex]\(102\)[/tex] is the remainder.
5. The numbers above the remainder [tex]\(3, 13, 32\)[/tex] are the coefficients of the quotient polynomial. Therefore, the quotient polynomial is [tex]\(3x^2 + 13x + 32\)[/tex] and the remainder is [tex]\(102\)[/tex].
In conclusion,
[tex]\[
\frac{3x^3 + 4x^2 - 7x + 6}{x - 3} = 3x^2 + 13x + 32 + \frac{102}{x - 3}
\][/tex]
