To divide the polynomial \( 3x^3 + 17x^2 + 21x - 9 \) by \( x + 3 \) using synthetic division, we follow these steps:
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Identify the value to use for synthetic division: We set \( x + 3 = 0 \), thus \( x = -3 \).
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Set up the coefficients of the polynomial: The coefficients for \( 3x^3 + 17x^2 + 21x - 9 \) are \( 3, 17, 21, -9 \).
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Perform synthetic division:
- Write down \( -3 \) (since we are dividing by \( x + 3 \)) and the coefficients:
\[ \begin{array}{r|rrrr} -3 & 3 & 17 & 21 & -9 \ & & -9 & -24 & 9 \ \hline & 3 & 8 & -3 & 0 \ \end{array} \]
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Bring down the first coefficient \( 3 \).
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Multiply \( 3 \) by \( -3 \) (the number on the left), which gives \( -9 \). Add this to the next coefficient \( 17 \):
\( 17 + (-9) = 8 \)
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Now multiply \( 8 \) by \( -3 \):
\( 8 \times -3 = -24 \). Add this to \( 21 \):
\( 21 + (-24) = -3 \)
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Finally, multiply \( -3 \) by \( -3 \):
\( -3 \times -3 = 9 \). Add this to \( -9 \):
\( -9 + 9 = 0 \)
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Write the result: The result of the synthetic division provides us with the new coefficients, which represent the quotient polynomial and a remainder.
The quotient is \( 3x^2 + 8x - 3 \) and the remainder is \( 0 \).
So the final answer is:
\[ 3x^2 + 8x - 3 \]
Thus, the correct response is:
3x² + 8x - 3.