To use synthetic division to divide the polynomial \( x^5 - 9x^3 + 2x \) by \( x + 4 \), we first identify the divisor, which in this case is \( -4 \) (since \( x + 4 \) is equivalent to setting \( x = -4 \)).
We then write down the coefficients of the polynomial. Given \( x^5 - 9x^3 + 2x \), there are missing coefficients for some terms:
- \( x^5 \) has a coefficient of 1.
- \( x^4 \) is missing, so its coefficient is 0.
- \( x^3 \) has a coefficient of -9.
- \( x^2 \) is missing, so its coefficient is 0.
- \( x \) has a coefficient of 2.
- The constant term is missing, so its coefficient is 0.
Thus, the coefficients to work with are \( [1, 0, -9, 0, 2, 0] \).
### Step-by-Step Synthetic Division:
1. Write down the coefficients: \( 1, 0, -9, 0, 2, 0 \).
2. Write the divisor: \( -4 \).
Start synthetic division process by bringing down the leading coefficient (1):
[tex]\[
\begin{array}{r|rrrrrr}
-4 & 1 & 0 & -9 & 0 & 2 & 0 \\
& & (1 \times -4) & (-4 \times -4) & (7 \times -4) & (-28 \times -4) & (114 \times -4)\\
\hline
& 1 & -4 & 7 & -28 & 114 & 112 \\
\end{array}
\][/tex]
Perform the steps one by one:
- Bring down the first coefficient, 1.
- Multiply 1 by -4 and add to the next coefficient: \( 0 + (-4) = -4 \).
- Multiply -4 by -4 and add to the next coefficient: \( -9 + 16 = 7 \).
- Multiply 7 by -4 and add to the next coefficient: \( 0 + (-28) = -28 \).
- Multiply -28 by -4 and add to the next coefficient: \( 2 + 112 = 114 \).
- Multiply 114 by -4 and add to the next coefficient: \( 0 - 456 = -456 \).
- The final number, 112, is the remainder.
So, the quotient coefficients are \( [1, -4, 7, -28, 114] \) and the remainder is 112.
Thus,
Quotient: \( x^4 - 4x^3 + 7x^2 - 28x + 114 \)
Remainder: [tex]\( 112 \)[/tex]
