To perform synthetic division with the polynomial \(2x^3 - 8x^2 - 19x - 30\) by the binomial \(x - 6\), we first need to use the zero of the divisor, which is \(x = 6\).
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Write down the coefficients of the polynomial:
- The coefficients for \(2x^3 - 8x^2 - 19x - 30\) are \(2, -8, -19, -30\).
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Set up synthetic division: \[ \begin{array}{r|rrrr} 6 & 2 & -8 & -19 & -30 \ & & 12 & 24 & 30 \ \hline & 2 & 4 & 5 & 0 \ \end{array} \]
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Perform synthetic division steps:
- Bring down the first coefficient: \(2\).
- Multiply \(6\) by \(2\) (the value below the line) to get \(12\) and add it to \(-8\): \(-8 + 12 = 4\).
- Next, multiply \(6\) by \(4\) to get \(24\) and add it to \(-19\): \(-19 + 24 = 5\).
- Now multiply \(6\) by \(5\) to get \(30\) and add it to \(-30\): \(-30 + 30 = 0\).
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The result of the synthetic division gives us: \[ 2x^2 + 4x + 5 \] with a remainder of \(0\).
Thus, when \(2x^3 - 8x^2 - 19x - 30\) is divided by \(x - 6\), the result is: \[ 2x^2 + 4x + 5 \]
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