Graph the polynomial function by first factoring and then using the steps for analyzing the graph of a polynomial function.
f left parenthesis x right parenthesis equals x cubed plus 4 x squared minus 7 x minus 10 Factor the polynomial.

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):
\[
\begin{array}{r|rrrr}
-2 & 1 & 4 & -7 & -10 \\
   &   & -2 & -4 & 22 \\
\hline
   & 1 & 2  & -11 & 12
\end{array}
\]
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.

Graph the polynomial function by first factoring and then using the steps for analyzing the graph of a polynomial function. f(x) = x^3 + 4x^2 - 7x - 10 Factor the polynomial.

Learn how to graph and factor the cubic polynomial f(x) = x^3 + 4x^2 - 7x - 10. Follow a detailed step-by-step approach including using the Rational Root Theorem, analyzing zeros, end behavior, and multiplicity. Suitable for high school students in Grades 10-12.

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