To decompose the polynomial \(10x^3 + 11x^2 - 27x - 18\) using synthetic division by the factor \((x + 2)\), we first set \(x + 2 = 0\), which gives us \(x = -2\).
Next, we can perform synthetic division using \(-2\), and the coefficients of the polynomial \(10, 11, -27, -18\):
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Write down the coefficients: \[ \begin{array}{r|rrrr} -2 & 10 & 11 & -27 & -18 \ & & & & \ \end{array} \]
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Bring down the leading coefficient (10): \[ \begin{array}{r|rrrr} -2 & 10 & 11 & -27 & -18 \ & & & & \ & 10 & & & \ \end{array} \]
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Multiply \(-2\) by \(10\) and add to the next coefficient (11): \(-2 \times 10 = -20\), and \(11 + (-20) = -9\): \[ \begin{array}{r|rrrr} -2 & 10 & 11 & -27 & -18 \ & & -20 & & \ & 10 & -9 & & \ \end{array} \]
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Multiply \(-2\) by \(-9\) and add to \(-27\): \(-2 \times -9 = 18\), and \(-27 + 18 = -9\): \[ \begin{array}{r|rrrr} -2 & 10 & 11 & -27 & -18 \ & & -20 & 18 & \ & 10 & -9 & -9 & \ \end{array} \]
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Multiply \(-2\) by \(-9\) and add to \(-18\): \(-2 \times -9 = 18\), and \(-18 + 18 = 0\): \[ \begin{array}{r|rrrr} -2 & 10 & 11 & -27 & -18 \ & & -20 & 18 & 0 \ & 10 & -9 & -9 & 0 \ \end{array} \]
The result of the synthetic division is: \[ 10x^2 - 9x - 9 \] Since the remainder is \(0\), we can confirm that \(x + 2\) is a factor of the original polynomial.
Next, we can factor the quadratic \(10x^2 - 9x - 9\). To factor this quadratic, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 10\), \(b = -9\), and \(c = -9\).
Calculating the discriminant: \[ b^2 - 4ac = (-9)^2 - 4 \cdot 10 \cdot (-9) = 81 + 360 = 441 \]
Now calculating the roots: \[ x = \frac{9 \pm \sqrt{441}}{20} = \frac{9 \pm 21}{20} \] Calculating the two roots:
- \(x = \frac{30}{20} = \frac{3}{2}\)
- \(x = \frac{-12}{20} = -\frac{3}{5}\)
Thus, we can factor the quadratic as: \[ 10x^2 - 9x - 9 = 10(x - \frac{3}{2})(x + \frac{3}{5}) \]
However, for simplicity, we can also write it as: \[ 10(x - \frac{3}{2})(x + \frac{3}{5}) = (x + 2)(10x^2 - 9x - 9) \]
The complete factorization can be written as: \[ 10(x + 2)\left(x - \frac{3}{2}\right)\left(x + \frac{3}{5}\right) \]
Summarizing, the factors of \(10x^3 + 11x^2 - 27x - 18\) are: \[ (x + 2)(10x^2 - 9x - 9) \]