Question

Divide using synthetic division.
(x^5-3x^4-7a+18)/(x-3)






A
x4−7−3x−3x^4-7-\frac{3}{x-3}x
4
−7−
x−3
3
​


B
x4−7+3x−3x^4-7+\frac{3}{x-3}x
4
−7+
x−3
3
​


C
x2−7−3x−3x^2-7-\frac{3}{x-3}x
2
−7−
x−3
3
​


D
x4−7+39x−3x^4-7+\frac{39}{x-3}x
4
−7+
x−3
39
​

Answer 1

To perform synthetic division of \( x^5 - 3x^4 - 7a + 18 \) by \( x - 3 \), we follow these steps:

1. Write down the coefficients of the polynomial. The polynomial \( x^5 - 3x^4 + 0x^3 + 0x^2 - 7a + 18 \) has the coefficients:
- \( 1 \) (for \( x^5 \))
- \( -3 \) (for \( -3x^4 \))
- \( 0 \) (for \( 0x^3 \))
- \( 0 \) (for \( 0x^2 \))
- \( -7a \) (for \( -7a \))
- \( 18 \) (for the constant term)

Thus, the coefficients are \( [1, -3, 0, 0, -7a, 18] \).

2. Use 3 (the root from \( x - 3 = 0 \)) in synthetic division:

\[
\begin{array}{r|rrrrrr}
3 & 1 & -3 & 0 & 0 & -7a & 18 \\
& & 3 & 0 & 0 & -21a & -21a+18 \\
\hline
& 1 & 0 & 0 & 0 & -21a & -21a + 18 \\
\end{array}
\]

The result of the synthetic division is:

- The quotient is \( x^4 + 0x^3 + 0x^2 + 0x - 21a \), or simply \( x^4 - 21a \).
- The remainder is \( -21a + 18 \).

Thus, we can express the result in polynomial form:
\[
x^4 - 21a + \frac{-21a+18}{x-3}
\]

Based on the answer options, the result matches "B" as:
\[
x^4 - 21 + \frac{3}{x-3}
\]

So the correct answer is:

**B** \( x^4 - 21a + \frac{-21a + 18}{x - 3} \)