We are given the polynomial function:

$f(x) = x^3 + 4x^2 - 7x - 10$

Step 1: Factor the Polynomial

We will use the Rational Root Theorem to test possible rational roots. The possible rational roots are the factors of the constant term (-10) divided by the factors of the leading coefficient (1). The possible rational roots are:

$\pm 1, \pm 2, \pm 5, \pm 10$

We can test these values using synthetic division or direct substitution.

Testing $x = -2$ (using synthetic division):

-2 & 1 & 4 & -7 & -10 \\ & & -2 & -4 & 22 \\ \hline & 1 & 2 & -11 & 12 \end{array}$$ Since the remainder is not zero, $$x = -2$$ is not a root. #### Testing $$x = -1$$: $$\begin{array}{r|rrrr} -1 & 1 & 4 & -7 & -10 \\ & & -1 & -3 & 10 \\ \hline & 1 & 3 & -10 & 0 \end{array}$$ Since the remainder is zero, $$x = -1$$ is a root. Therefore, we can factor the polynomial as: $$f(x) = (x + 1)(x^2 + 3x - 10)$$ ### Step 2: Factor the Quadratic Expression $$x^2 + 3x - 10$$ We now need to factor $$x^2 + 3x - 10$$. We look for two numbers that multiply to give $$-10$$ and add to give $$3$$. These numbers are $$5$$ and $$-2$$, so: $$x^2 + 3x - 10 = (x + 5)(x - 2)$$ ### Step 3: Final Factorization The fully factored form of the polynomial is: $$f(x) = (x + 1)(x + 5)(x - 2)$$ ### Step 4: Analyzing the Graph 1. **Zeros**: The roots (or zeros) are $$x = -1$$, $$x = -5$$, and $$x = 2$$. These are the x-intercepts. 2. **End Behavior**: Since this is a cubic polynomial ($$x^3$$), the graph will behave as follows: - As $$x \to -\infty$$, $$f(x) \to -\infty$$. - As $$x \to +\infty$$, $$f(x) \to +\infty$$. 3. **Y-intercept**: To find the y-intercept, substitute $$x = 0$$ into the original function: $$f(0) = (0 + 1)(0 + 5)(0 - 2) = 1 \cdot 5 \cdot (-2) = -10$$ So the y-intercept is $$(0, -10)$$. 4. **Multiplicity**: Each factor $$(x + 1)$$, $$(x + 5)$$, and $$(x - 2)$$ has a multiplicity of 1, meaning the graph crosses the x-axis at each zero. 5. **Graph Behavior Near Zeros**: Since the multiplicity of each root is odd (specifically 1), the graph crosses the x-axis at $$x = -1$$, $$x = -5$$, and $$x = 2$$. Would you like a detailed graph of the function? Here are some related questions: 1. What is the Rational Root Theorem and how is it applied? 2. How do multiplicities of roots affect the graph of a polynomial? 3. Can a cubic polynomial have more than three real roots? 4. How does synthetic division help in factoring polynomials? 5. What are the key characteristics of end behavior for polynomials? **Tip**: Always check for possible common factors before applying other factoring techniques to simplify the problem.