Solving a 3rd degree polynomial using Gauss's method

Problem statement

Given the following information about a third-degree polynomial function:

• The function passes through the x-axis at -1.5.
• The function has a high point at (-0.5, 4.5).
• The function has a point of inflection at x = 2/3.

We need to find the equation of the function.

Gauss's method with 4 unknowns

To find the equation of the function, we can use Gauss's method with 4 unknowns. The steps are as follows:

1. Write the function in the form f(x) = ax^3 + bx^2 + cx + d.
2. Use the fact that the function passes through the x-axis at -1.5 to form an equation involving d.
3. Use the fact that the function has a high point at (-0.5, 4.5) to form two equations involving a, b, and c.
4. Use the fact that the function has a point of inflection at x = 2/3 to form an equation involving a and b.
5. Solve the system of equations to find the values of a, b, c, and d.
6. Substitute the values of a, b, c, and d back into the equation f(x) = ax^3 + bx^2 + cx + d to get the final equation.

Step 1: Write the function in the form f(x) = ax^3 + bx^2 + cx + d

Let's assume the function has the form f(x) = ax^3 + bx^2 + cx + d.

Step 2: Use the fact that the function passes through the x-axis at -1.5 to form an equation involving d.

Since the function passes through the x-axis at -1.5, we know that f(-1.5) = 0. Using our assumed form of the function, we get:

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a(-1.5)^3 + b(-1.5)^2 + c(-1.5) + d = 0
-3.375a + 2.25b - 1.5c + d = 0

This gives us our first equation.

Step 3: Use the fact that the function has a high point at (-0.5, 4.5) to form two equations involving a, b, and c.

Since the function has a high point at (-0.5, 4.5), we know that the derivative of the function is zero at x = -0.5. Taking the derivative of our assumed form of the function, we get:

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f'(x) = 3ax^2 + 2bx + c

Setting x = -0.5 and f'(-0.5) = 0, we get:

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3a(-0.5)^2 + 2b(-0.5) + c = 0
0.75a - b + c = 0   (Equation 1)

Also, plugging in (-0.5, 4.5) into our assumed form of the function, we get:

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a(-0.5)^3 + b(-0.5)^2 + c(-0.5) + d = 4.5
-0.125a + 0.25b - 0.5c + d = 4.5   (Equation 2)

These two equations give us the information needed to solve for a, b, and c.

Step 4: Use the fact that the function has