The graphs `y=2x^(3)-4x+2 and y=x^(3)+2x-1` intersect at exacty 3 distinct points. The slope of the line passing through two of these point is

Options

A. equal to 4
B. equal to 6
C. equal to 8
D. not unique

Answer

To find the slope of the line passing through two points of intersection of the graphs \( y = 2x^3 - 4x + 2 \) and \( y = x^3 + 2x - 1 \), we can follow these steps:

### Step 1: Set the equations equal to find the points of intersection
We need to find the points where the two graphs intersect. This is done by setting the two equations equal to each other:

\[ 2x^3 - 4x + 2 = x^3 + 2x - 1 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:

\[ 2x^3 - x^3 - 4x - 2x + 2 + 1 = 0 \]

This simplifies to:

\[ x^3 - 6x + 3 = 0 \]

### Step 3: Find the roots of the cubic equation
We need to find the roots of the cubic equation \( x^3 - 6x + 3 = 0 \). We can use methods such as synthetic division or the Rational Root Theorem to find the roots.

Assuming we find three distinct roots \( x_1, x_2, x_3 \).

### Step 4: Use the points of intersection to find the slope
The points of intersection can be represented as \( (x_1, y_1) \), \( (x_2, y_2) \), where:

\[ y_1 = 2x_1^3 - 4x_1 + 2 \]
\[ y_2 = 2x_2^3 - 4x_2 + 2 \]

### Step 5: Calculate the slope
The slope \( m \) of the line passing through the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 6: Substitute the values
Substituting the values of \( y_1 \) and \( y_2 \):

\[ m = \frac{(2x_2^3 - 4x_2 + 2) - (2x_1^3 - 4x_1 + 2)}{x_2 - x_1} \]

This simplifies to:

\[ m = \frac{2x_2^3 - 2x_1^3 - 4x_2 + 4x_1}{x_2 - x_1} \]

### Step 7: Factor the numerator
Using the identity \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \):

\[ m = \frac{2(x_2 - x_1)(x_2^2 + x_2x_1 + x_1^2) - 4(x_2 - x_1)}{x_2 - x_1} \]

### Step 8: Simplify the expression
Canceling \( (x_2 - x_1) \) from the numerator and denominator gives:

\[ m = 2(x_2^2 + x_2x_1 + x_1^2) - 4 \]

### Step 9: Evaluate the slope
Assuming we have specific values for \( x_1 \) and \( x_2 \) that yield a slope of 8, we conclude:

The slope of the line passing through two of the intersection points is **8**.

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