Bot GPT 4-o mini To perform synthetic division of \( x^5 - 3x^4 - 7a + 18 \) by \( x - 3 \), we follow these steps:
- 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 \( [1, -3, 0, 0, -7a, 18] \).
- 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 \( x^4 - 21a + \frac{-21a + 18}{x - 3} \)
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