0=(0.2x^3-0.3x^2 0.1)(-0.1x^5 0.4x^2 4) 0.3

General
1

0=(0.2x^3-0.3x^2 0.1)(-0.1x^5 0.4x^2 4) 0.3

2

How do you solve the equation (0 = (0.2x^3 - 0.3x^2 + 0.1)(-0.1x^5 + 0.4x^2 + 4) + 0.3)?

Answer:

To solve this equation, you’ll need to follow these steps: distribute the factors, simplify the expression, and solve for (x). Let’s break down the problem methodically:

Step 1: Expand the Expression

Given:

0 = (0.2x^3 - 0.3x^2 + 0.1)(-0.1x^5 + 0.4x^2 + 4) + 0.3

First, let’s expand the expression by distributing each term from the first polynomial to each term from the second polynomial:

  1. Distribute (0.2x^3):

    [
    (0.2x^3)(-0.1x^5) + (0.2x^3)(0.4x^2) + (0.2x^3)(4)
    ]

    • (0.2 \times -0.1 = -0.02), and the powers multiplied give (x^{3+5} = x^8): (-0.02x^8)
    • (0.2 \times 0.4 = 0.08), and the powers multiplied give (x^{3+2} = x^5): (0.08x^5)
    • (0.2 \times 4 = 0.8): (0.8x^3)

  2. Distribute (-0.3x^2):

    [
    (-0.3x^2)(-0.1x^5) + (-0.3x^2)(0.4x^2) + (-0.3x^2)(4)
    ]

    • (-0.3 \times -0.1 = 0.03), and the powers multiplied give (x^{2+5} = x^7): (0.03x^7)
    • (-0.3 \times 0.4 = -0.12), and the powers multiplied give (x^{2+2} = x^4): (-0.12x^4)
    • (-0.3 \times 4 = -1.2): (-1.2x^2)

  3. Distribute (0.1):

    [
    (0.1)(-0.1x^5) + (0.1)(0.4x^2) + (0.1)(4)
    ]

    • (0.1 \times -0.1 = -0.01): (-0.01x^5)
    • (0.1 \times 0.4 = 0.04): (0.04x^2)
    • (0.1 \times 4 = 0.4)

Step 2: Combine Like Terms

Now we combine all the like terms from the expanded expression:

  • The terms are: (-0.02x^8, 0.03x^7, (0.08 - 0.01)x^5, -0.12x^4, (-1.2 + 0.04)x^2, 0.8, 0.4)

Combine them:

  • (x^8): (-0.02x^8)
  • (x^7): (0.03x^7)
  • (x^5): (0.08x^5 - 0.01x^5 = 0.07x^5)
  • (x^4): (-0.12x^4)
  • (x^2): (-1.2x^2 + 0.04x^2 = -1.16x^2)
  • Constant: (0.8 + 0.4 = 1.2)

The equation becomes:

0 = -0.02x^8 + 0.03x^7 + 0.07x^5 - 0.12x^4 - 1.16x^2 + 1.2 + 0.3

Combine the constants:

1.2 + 0.3 = 1.5

So, the simplified equation is:

0 = -0.02x^8 + 0.03x^7 + 0.07x^5 - 0.12x^4 - 1.16x^2 + 1.5

Step 3: Solve for (x)

This equation is a high-degree polynomial and may not have straightforward analytical solutions. You may need to use numerical methods or graphing techniques to find solutions:

  1. Graphical Approach:

    • Graph the polynomial and look for x-intercepts, which represent the roots of the equation.

  2. Numerical Methods:

    • Use calculus-based methods like Newton-Raphson, or digital tools such as a graphing calculator or software like Desmos, Mathematica, or MATLAB to approximate the solutions.

Solving the equation exactly requires techniques beyond simple algebra due to its complexity (e.g., degree 8). However, using the methods above, you should be able to find approximate solutions.

If you have specific values or constraints in mind, let me know to provide further assistance! @anonymous13