To perform the polynomial division of [tex]\( \frac{4x^3 + 6x + 18}{x + 2} \)[/tex], we will follow a step-by-step process.
1. Set up the division:
Write [tex]\(4x^3 + 6x + 18\)[/tex] under the long division bar, and [tex]\(x + 2\)[/tex] outside the bar.
[tex]\[
\begin{array}{r|lll}
& 4x^3 + 0x^2 + 6x + 18 \\
x + 2 & \\
\end{array}
\][/tex]
2. First term:
Divide the leading term of the numerator [tex]\(4x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex].
[tex]\[
\frac{4x^3}{x} = 4x^2
\][/tex]
Write [tex]\(4x^2\)[/tex] as the first term of the quotient.
[tex]\[
\begin{array}{r|lll}
& 4x^2 \\
x + 2 & 4x^3 + 0x^2 + 6x + 18 \\
\end{array}
\][/tex]
3. Multiply and subtract:
Multiply [tex]\(4x^2\)[/tex] by [tex]\(x + 2\)[/tex], and subtract the result from the current polynomial.
[tex]\[
4x^2 \cdot (x + 2) = 4x^3 + 8x^2
\][/tex]
[tex]\[
\begin{array}{r|lll}
& 4x^2 \\
x + 2 & 4x^3 + 0x^2 + 6x + 18 \\
& -(4x^3 + 8x^2) \\
\hline
& -8x^2 + 6x + 18 \\
\end{array}
\][/tex]
4. Second term:
Next, divide the new leading term [tex]\(-8x^2\)[/tex] by the leading term [tex]\(x\)[/tex].
[tex]\[
\frac{-8x^2}{x} = -8x
\][/tex]
Write [tex]\(-8x\)[/tex] as the next term of the quotient.
[tex]\[
\begin{array}{r|lll}
& 4x^2 - 8x \\
x + 2 & 4x^3 + 0x^2 + 6x + 18 \\
& -(4x^3 + 8x^2) \\
\hline
& -8x^2 + 6x + 18 \\
\end{array}
\][/tex]
5. Multiply and subtract:
Multiply [tex]\(-8x\)[/tex] by [tex]\(x + 2\)[/tex], and subtract the result from the current polynomial.
[tex]\[
-8x \cdot (x + 2) = -8x^2 - 16x
\][/tex]
[tex]\[
\begin{array}{r|lll}
& 4x^2 - 8x \\
x + 2 & 4x^3 + 0x^2 + 6x + 18 \\
& -(4x^3 + 8x^2) \\
\hline
& -8x^2 + 6x + 18 \\
& -(-8x^2 - 16x) \\
\hline
& 22x + 18 \\
\end{array}
\][/tex]
6. Third term:
Finally, divide the new leading term [tex]\(22x\)[/tex] by the leading term [tex]\(x\)[/tex].
[tex]\[
\frac{22x}{x} = 22
\][/tex]
Write [tex]\(22\)[/tex] as the last term of the quotient.
[tex]\[
\begin{array}{r|lll}
& 4x^2 - 8x + 22 \\
x + 2 & 4x^3 + 0x^2 + 6x + 18 \\
& -(4x^3 + 8x^2) \\
\hline
& -8x^2 + 6x + 18 \\
& -(-8x^2 - 16x) \\
\hline
& 22x + 18 \\
& -(22x + 44) \\
\hline
& -26 \\
\end{array}
\][/tex]
Thus, the quotient is [tex]\( 4x^2 - 8x + 22 \)[/tex] and the remainder is [tex]\(-26\)[/tex].
So,
[tex]\[
\frac{4x^3 + 6x + 18}{x + 2} = 4x^2 - 8x + 22 \quad \text{with a remainder of} \quad -26
\][/tex]
