Math Problem Statement

Find the zeros and analyze the end behavior of the function f(x) = x^3 + 2x^2 - 5x - 6.

Solution

To solve the problem based on the uploaded image:

We are tasked with analyzing the function f(x)=x3+2x2−5x−6f(x) = x^3 + 2x^2 - 5x - 6. Here's the process:

1. Finding the Zeros of f(x)f(x):

The zeros of the function occur where f(x)=0f(x) = 0. This requires solving the cubic equation:

x3+2x2−5x−6=0.x^3 + 2x^2 - 5x - 6 = 0.

To solve this, we use the Rational Root Theorem to test potential rational roots (factors of the constant term −6-6 over the leading coefficient 11). Possible roots are:

±1,±2,±3,±6.\pm 1, \pm 2, \pm 3, \pm 6.

Testing these values:

  • f(−3)=(−3)3+2(−3)2−5(−3)−6=−27+18+15−6=0f(-3) = (-3)^3 + 2(-3)^2 - 5(-3) - 6 = -27 + 18 + 15 - 6 = 0, so x=−3x = -3 is a root.

Using synthetic division to divide f(x)f(x) by (x+3)(x + 3):

-3 & 1 & 2 & -5 & -6 \\ & & -3 & 3 & 6 \\ \hline & 1 & -1 & -2 & 0 \\ \end{array}$$ This gives the quotient $$x^2 - x - 2$$. Factoring $$x^2 - x - 2$$: $$x^2 - x - 2 = (x - 2)(x + 1).$$ Therefore, the roots of $$f(x)$$ are: $$x = -3, \, x = -1, \, x = 2.$$ ### 2. **End Behavior of $$f(x)$$:** The end behavior of a cubic function depends on the leading term $$x^3$$. As $$x \to \infty$$ or $$x \to -\infty$$: - As $$x \to \infty$$, $$f(x) \to \infty$$. - As $$x \to -\infty$$, $$f(x) \to -\infty$$. ### 3. **Correct Selections:** - **Zeros:** The correct points on the graph are at $$x = -3$$, $$x = -1$$, and $$x = 2$$. - **End Behavior:** - As $$x \to \infty$$, $$f(x) \to \infty$$. - As $$x \to -\infty$$, $$f(x) \to -\infty$$. Would you like to see a step-by-step explanation of the synthetic division or a visualization of the function's graph? --- ### Related Questions: 1. What is the significance of the Rational Root Theorem in solving polynomial equations? 2. How does the degree of a polynomial affect its end behavior? 3. Could $$f(x)$$ have additional complex roots? Why or why not? 4. How do we interpret the derivative $$f'(x)$$ for determining turning points of $$f(x)$$? 5. What would change in the solution if the leading term was negative? --- ### Tip: For cubic functions, always verify one root before simplifying to a quadratic equation—it simplifies the solving process greatly.

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Math Problem Analysis

Mathematical Concepts

Cubic Functions
Synthetic Division
Rational Root Theorem
Factoring

Formulas

x^3 + 2x^2 - 5x - 6 = 0
f(x) = x^3 + 2x^2 - 5x - 6

Theorems

Rational Root Theorem
Synthetic Division

Suitable Grade Level

Grades 9-12

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