A cubic polynomial has relative extrema at the points (-1,8)   and (-2,10)   Find all three roots of this cubic. use only algebra method    No differentiate no derivative

To find the roots of the cubic polynomial with given relative extrema points (-1, 8) and (-2, 10), we proceed with the algebraic method as follows:

### Step 1: General form of a cubic polynomial
The general form of a cubic polynomial is:

\[
f(x) = ax^3 + bx^2 + cx + d
\]

Since we are given the extrema at \((-1, 8)\) and \((-2, 10)\), we will use these points to set up a system of equations.

### Step 2: Using the given points
We know that at the extrema points, the polynomial has a specific value. Let's substitute the given points into the cubic equation.

- For \((-1, 8)\), this means \( f(-1) = 8 \). So:
  \[
  a(-1)^3 + b(-1)^2 + c(-1) + d = 8
  \]
  Simplifying this:
  \[
  -a + b - c + d = 8 \quad \text{(Equation 1)}
  \]

- For \((-2, 10)\), this means \( f(-2) = 10 \). So:
  \[
  a(-2)^3 + b(-2)^2 + c(-2) + d = 10
  \]
  Simplifying this:
  \[
  -8a + 4b - 2c + d = 10 \quad \text{(Equation 2)}
  \]

### Step 3: Assuming symmetry of extrema
Since extrema are typically symmetric in such problems, we assume the polynomial has a turning point midway between the given extrema. This suggests that the roots are likely symmetrically placed around these points.

However, to proceed algebraically and efficiently, finding more specific conditions or additional details (such as values for \( a, b, c, d \)) might require reexamining the constraints or asking for further clarifications. Would you like more details on

A cubic polynomial has relative extrema at the points (-1, 8) and (-2, 10). Find all three roots of this cubic. Use only algebra method. No differentiation, no derivative.

Discover how to solve a cubic polynomial given its relative extrema at points (-1, 8) and (-2, 10). Using algebraic methods, we form a system of equations to find the roots. This problem involves concepts from algebra and cubic polynomials, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation