Questions: Graph the polynomial function. h(x)=x^3+6x^2-x-6

Transcript text: Graph the polynomial function. \[ h(x)=x^{3}+6 x^{2}-x-6 \]

Solution

Solution Steps

Step 1: Find the roots of the polynomial

To find the roots of the polynomial $ h(x) = x^3 + 6x^2 - x - 6 $, we can use the Rational Root Theorem to test possible rational roots. The possible rational roots are the factors of the constant term, $-6$, which are $\pm 1, \pm 2, \pm 3, \pm 6$.

Testing these values, we find that $ x = -3 $ is a root of the polynomial.

Step 2: Factor the polynomial

Since $ x = -3 $ is a root, we can factor the polynomial as $ h(x) = (x + 3)(x^2 + 3x - 2) $.

Step 3: Solve the quadratic equation

Now, solve the quadratic equation $ x^2 + 3x - 2 = 0 $ using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where $ a = 1 $, $ b = 3 $, and $ c = -2 $.

\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{-3 \pm \sqrt{9 + 8}}{2} = \frac{-3 \pm \sqrt{17}}{2} \]

Thus, the roots of the polynomial are $ x = -3 $, $ x = \frac{-3 + \sqrt{17}}{2} $, and $ x = \frac{-3 - \sqrt{17}}{2} $.

Final Answer

The roots of the polynomial $ h(x) = x^3 + 6x^2 - x - 6 $ are $ x = -3 $, $ x = \frac{-3 + \sqrt{17}}{2} $, and $ x = \frac{-3 - \sqrt{17}}{2} $.

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 2, "ymin": -20, "ymax": 20}, "commands": ["y = x3 + 6*x2 - x - 6"], "latex_expressions": ["$y = x^3 + 6x^2 - x - 6$"]}

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