Sure, let's perform synthetic division step by step to divide the polynomial [tex]\(4x^3 + 8x^2 - 4x + 9\)[/tex] by [tex]\(x + 3\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\(4x^3 + 8x^2 - 4x + 9\)[/tex] has coefficients 4, 8, -4, and 9.
2. Find the zero of the divisor:
The divisor is [tex]\(x + 3\)[/tex], which means [tex]\(x = -3\)[/tex] is the zero of the divisor.
3. Set up the synthetic division:
We will write the coefficients in a row: [tex]\(4, 8, -4, 9\)[/tex].
We write the zero of the divisor, which is [tex]\(-3\)[/tex], to the left.
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
\end{array}
\][/tex]
4. Perform the synthetic division steps:
- Start by bringing down the first coefficient (4) directly below the line.
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & & & \\
\end{array}
\][/tex]
- Multiply the zero [tex]\(-3\)[/tex] by the number just written below the line, and write the result below the next coefficient (8).
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & -12 & & \\
\end{array}
\][/tex]
- Add this result to the next coefficient (8), and write the sum below the line.
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & -4 & & \\
\end{array}
\][/tex]
- Repeat the process: multiply [tex]\(-3\)[/tex] by the latest number written below the line ([tex]\(-4\)[/tex]), and write the result below the next coefficient ([tex]\(-4\)[/tex]).
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & -4 & 12 & \\
\end{array}
\][/tex]
- Add this to the next coefficient ([tex]\(-4\)[/tex]), and write the sum below the line.
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & -4 & 8 & \\
\end{array}
\][/tex]
- Repeat the process one more time: multiply [tex]\(-3\)[/tex] by the latest number written below the line (8), and write the result below the next coefficient (9).
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & -4 & 8 & -24 \\
\end{array}
\][/tex]
- Add this to the next coefficient (9), and write the sum below the line.
[tex]\[
\begin{array}{r|rrrr}
-3 & 4 & 8 & -4 & 9 \\
& & 4 & -4 & 8 & -15 \\
\end{array}
\][/tex]
5. Identify the quotient and the remainder:
- The numbers below the line, excluding the very last one, give us the coefficients of the quotient polynomial.
- The last number is the remainder.
So, the quotient is [tex]\(4x^2 - 4x + 8\)[/tex] and the remainder is [tex]\(-15\)[/tex].
Therefore, the final answers are:
- The quotient is [tex]\(Q(x) = 4x^2 - 4x + 8\)[/tex].
- The remainder is [tex]\(R(x) = -15\)[/tex].
