To divide the polynomial [tex]\(x^2 + 2x + 6\)[/tex] by [tex]\(x - 3\)[/tex], we will perform polynomial long division step-by-step.
1. Set up the division:
[tex]\[
\begin{array}{r|l}
x - 3 & x^2 + 2x + 6 \\
\end{array}
\][/tex]
2. Divide the first term of the numerator by the first term of the divisor:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
- Write [tex]\(x\)[/tex] above the division line:
[tex]\[
\begin{array}{r|ll}
x - 3 & x^2 + 2x + 6 \\
& x \\
\end{array}
\][/tex]
3. Multiply the entire divisor by this result:
[tex]\[
x \cdot (x - 3) = x^2 - 3x
\][/tex]
4. Subtract this product from the original polynomial:
[tex]\[
\begin{array}{r|ll}
x - 3 & x^2 + 2x + 6 \\
& x^2 - 3x \\
\hline
& 5x + 6 \\
\end{array}
\][/tex]
5. Bring down the next term (if any).
- Move down the [tex]\(5x + 6\)[/tex] completely.
6. Repeat the division with the new polynomial:
- Divide [tex]\(5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{5x}{x} = 5
\][/tex]
- Write [tex]\(5\)[/tex] above the division line, next to [tex]\(x\)[/tex]:
[tex]\[
\begin{array}{r|lll}
x - 3 & x^2 + 2x + 6 \\
& x + 5 \\
\end{array}
\][/tex]
7. Multiply the entire divisor by this result:
[tex]\[
5 \cdot (x - 3) = 5x - 15
\][/tex]
8. Subtract this product from the new polynomial:
[tex]\[
\begin{array}{r|lll}
x - 3 & x^2 + 2x + 6 \\
& x^2 - 3x \\
\hline
& 5x + 6 \\
& 5x - 15 \\
\hline
& 21 \\
\end{array}
\][/tex]
Rewriting the remainders and quotients, we get:
[tex]\[
x^2 + 2x + 6 = (x - 3)(x + 5) + 21
\][/tex]
So, the quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(21\)[/tex].
Therefore, [tex]\((x^2 + 2x + 6) \div (x - 3) = x + 5\)[/tex] with a remainder of [tex]\(21\)[/tex].
