To perform synthetic division of the polynomial \(2x^3 + 3x^2 - 21x + 20\) by \(x - 2\), we will use \(2\) (the root of \(x - 2 = 0\)) in synthetic division.
Here are the steps of the synthetic division:
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Write down the coefficients of the polynomial:
- For \(2x^3\), the coefficient is \(2\).
- For \(3x^2\), the coefficient is \(3\).
- For \(-21x\), the coefficient is \(-21\).
- For the constant term \(20\), the coefficient is \(20\).
So we have the coefficients: \(2, 3, -21, 20\).
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Set up for synthetic division:
\[ \begin{array}{r|rrrr} 2 & 2 & 3 & -21 & 20 \ & & 4 & 14 & -14 \ \hline & 2 & 7 & -7 & 6 \ \end{array} \]
- Bring down the \(2\).
- Multiply \(2\) by \(2\) (the value being used in synthetic division) and write \(4\) underneath the next coefficient \(3\).
- Add \(3 + 4 = 7\).
- Multiply \(2\) by \(7\), and write \(14\) underneath \(-21\).
- Add \(-21 + 14 = -7\).
- Multiply \(2\) by \(-7\), and write \(-14\) underneath \(20\).
- Add \(20 - 14 = 6\).
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The result of the synthetic division is:
- The coefficients of the quotient polynomial \(2x^2 + 7x - 7\).
- The remainder \(6\).
So, you can express the result of the division as:
\[ \frac{2x^3 + 3x^2 - 21x + 20}{x - 2} = 2x^2 + 7x - 7 + \frac{6}{x - 2}. \]
Thus, the final result is:
\[ q(x) = 2x^2 + 7x - 7, \quad r(x) = 6, \quad b(x) = x - 2. \]
Therefore, the complete expression is:
\[ 2x^2 + 7x - 7 + \frac{6}{x-2}. \]
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