The x intercept of y = x ^ 3 - 14x ^ 2 + bx - 182 is? (Enter the coordinates of the x intercept ) The y- intercept of y = x ^ 3 - 14x ^ 2 + bx - 182 is ? (Enter the coordinates of the y-intercept ) Solve f(x) = 0 Find all solutions . x =? ( If there is more than one solution, separate with commas)

Let's solve the given cubic equation \( y = x^3 - 14x^2 + bx - 182 \) to find the x-intercept, y-intercept, and the solutions for \( f(x) = 0 \).

### 1. Finding the x-intercept:
The x-intercept occurs where \( y = 0 \). So, we need to solve:
\[ x^3 - 14x^2 + bx - 182 = 0 \]

### 2. Finding the y-intercept:
The y-intercept occurs where \( x = 0 \). Plugging \( x = 0 \) into the equation gives:
\[ y = 0^3 - 14 \cdot 0^2 + b \cdot 0 - 182 = -182 \]
Therefore, the y-intercept is:
\[ (0, -182) \]

### 3. Solving \( f(x) = 0 \):
To find the solutions for \( f(x) = 0 \), we need the roots of the equation \( x^3 - 14x^2 + bx - 182 = 0 \). 

### Step-by-step Approach:
1. **Identify known intercepts and factors:** Since we don't have a specific value for \( b \), let's explore possible rational roots using the Rational Root Theorem, which states that any rational root, in this case, must be a factor of the constant term (-182) divided by the leading coefficient (1), i.e., factors of -182.

   Factors of 182: \( \pm 1, \pm 2, \pm 7, \pm 13, \pm 14, \pm 26, \pm 91, \pm 182 \)

2. **Test possible roots:** By substituting these values into the polynomial to check if they satisfy \( x^3 - 14x^2 + bx - 182 = 0 \).

Let’s assume a root, say \( x = a \), then the polynomial \( x^3 - 14x^2 + bx - 182 = 0 \) must be true for \( x = a \).

3. **Synthetic Division or Factoring:** To find the other roots, we can use synthetic division or factoring methods once we find one root.

Since the exact value of \( b \) is not given, we cannot find the exact solutions directly. Typically, we would use numerical methods or factorization techniques.

### Summary:
- **x-intercept(s):** (Dependent on finding the value of \( b \) and solving \( x^3 - 14x^2 + bx - 182 = 0 \))
- **y-intercept:** \((0, -182)\)
- **Solutions for \( f(x) = 0 \):** Dependent on the exact value of \( b \).

Please provide the value of \( b \) for further precise calculation.

Would you like detailed steps or have any questions about this process? 

Here are some relative questions to expand on this information:

1. How do you find the x-intercept of a polynomial function?
2. What is the Rational Root Theorem and how is it used?
3. How do you perform synthetic division?
4. How do you factor a cubic polynomial?
5. What are the possible rational roots for a given polynomial?
6. How can you verify if a given value is a root of a polynomial?
7. What methods can be used to solve a cubic equation if factoring is complex?
8. How does the value of \( b \) affect the roots of the cubic equation?

**Tip:** When solving cubic equations, always check for possible rational roots first to simplify the problem.

Learn how to solve the cubic equation y = x^3 - 14x^2 + bx - 182. Find the x-intercept, y-intercept, and all solutions using step-by-step methods. This problem covers cubic equations, polynomial intercepts, and root finding. Suitable for Grades 10-12.

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