Sure, let's go through the process of performing synthetic division step-by-step to divide \(5x^2 + 6x - 16\) by \(x + 4\).
Step 1: Identify the coefficients
The polynomial \(5x^2 + 6x - 16\) has coefficients \(5\), \(6\), and \(-16\).
The divisor \(x + 4\) can be rewritten in the form \(x - (-4)\). So, the root we use for synthetic division is \(-4\).
Step 2: Set up the synthetic division
Write the coefficients of the dividend in a row:
[tex]\[ 5, 6, -16 \][/tex]
Write the root used for synthetic division to the left:
[tex]\[ -4 \][/tex]
Step 3: Perform synthetic division
1. Bring down the first coefficient (5) directly below the line:
[tex]\[
\begin{array}{r|rrr}
-4 & 5 & 6 & -16 \\
& & & \\
\hline
& 5 & & \\
\end{array}
\][/tex]
2. Multiply the root (-4) by the value just written (5), and write the result under the next coefficient:
[tex]\[
-4 \times 5 = -20
\][/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 5 & 6 & -16 \\
& & -20 & \\
\hline
& 5 & -14 & \\
\end{array}
\][/tex]
3. Add this result to the next coefficient (6):
[tex]\[
6 + (-20) = -14
\][/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 5 & 6 & -16 \\
& & -20 & 56 \\
\hline
& 5 & -14 & 40 \\
\end{array}
\][/tex]
4. Multiply the root (-4) by the current value (-14) and write the result under the next coefficient:
[tex]\[
-4 \times -14 = 56
\][/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 5 & 6 & -16 \\
& & -20 & 56 \\
\hline
& 5 & -14 & 40 \\
\end{array}
\][/tex]
5. Add this result to the next coefficient (-16):
[tex]\[
-16 + 56 = 40
\][/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 5 & 6 & -16 \\
& & -20 & 56 \\
\hline
& 5 & -14 & 40 \\
\end{array}
\][/tex]
Step 4: Interpret the results
The final row represents the coefficients of the quotient and the remainder.
- The quotient is formed by the first two numbers: \(5x - 14\).
- The remainder is the last number: \(40\).
Step 5: Write the final answer
The division of \(5x^2 + 6x - 16\) by \(x + 4\) gives:
[tex]\[
5x^2 + 6x - 16 \div (x + 4) = 5x - 14 \text{ with a remainder of } 40
\][/tex]
So, the final answer simplifies to:
[tex]\[
5x^2 + 6x - 16 = (x + 4)(5x - 14) + 40
\][/tex]
