To find the zeros of the polynomial function \( g(x) = -2x^3 + 6x^2 - 32x + 96 \) using synthetic division, we can start by identifying potential rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (96) divided by the factors of the leading coefficient (-2).
The factors of 96 are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 32, \pm 48, \pm 96 \).
The factors of -2 are: \( \pm 1, \pm 2 \).
So the possible rational roots (p/q) that we will test are: \[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 32, \pm 48, \pm 96 \] divided by \( 1 \) and \( 2 \), which gives us: \[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 32, \pm 48, \pm 96, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{9}{2}, \pm 48 \]
Let's start testing some rational roots, beginning with \( x = 4 \):
Using synthetic division, set up as follows:
\[ \begin{array}{r|rrrr} 4 & -2 & 6 & -32 & 96 \ & & -8 & -8 & -32 \ \hline & -2 & -2 & -40 & 64 \ \end{array} \]
The final row gives us the coefficients of the resulting polynomial and the remainder: \(-2x^2 - 2x - 40.\)
Since the remainder is not zero, \( x = 4 \) is not a root. Now let's test \( x = 6 \).
Using synthetic division with \( x = 6\):
\[ \begin{array}{r|rrrr} 6 & -2 & 6 & -32 & 96 \ & & -12 & -36 & -48 \ \hline & -2 & -6 & -68 & 48 \ \end{array} \]
This also leads to a non-zero remainder. Next, we will test \( x = 8 \):
\[ \begin{array}{r|rrrr} 8 & -2 & 6 & -32 & 96 \ & & -16 & -80 & -96 \ \hline & -2 & -10 & -112 & 0 \ \end{array} \]
The last number is a zero, which implies \( x = 8 \) is a root!
Now we have the quadratic \( -2x^2 - 10x - 12 \) left. Let's simplify this:
Dividing the polynomial by -2: \[ x^2 + 5x + 6 \]
Now we can factor \( x^2 + 5x + 6 \): \[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Thus the roots from this factored representation are \( x = -2 \) and \( x = -3 \).
Summary of the roots:
The zeros of \( g(x) = -2x^3 + 6x^2 - 32x + 96 \) are: \[ x = 8, \quad x = -2, \quad x = -3. \]
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