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(4x+3)^2(2x+1)(x+1)=810

3 min read Jun 16, 2024
(4x+3)^2(2x+1)(x+1)=810

Solving the Equation (4x+3)^2(2x+1)(x+1) = 810

This equation presents a challenge because it involves a combination of polynomial terms. Here's a step-by-step approach to solve it:

1. Simplify the equation

  • Expand the squares: (4x+3)^2 = 16x^2 + 24x + 9
  • Rewrite the equation: (16x^2 + 24x + 9)(2x+1)(x+1) = 810

2. Expand the polynomial

  • Multiply the three factors together. This process involves carefully distributing each term.
  • Result: 32x^4 + 80x^3 + 70x^2 + 31x + 9 = 810

3. Convert to a standard form

  • Subtract 810 from both sides to get a standard polynomial equation: 32x^4 + 80x^3 + 70x^2 + 31x - 801 = 0

4. Finding the Roots

  • Factoring: This equation might be difficult to factor directly.
  • Rational Root Theorem: This theorem can help you find potential rational roots. It states that any rational root of the polynomial must be a divisor of the constant term (-801) divided by a divisor of the leading coefficient (32).
  • Numerical Methods: You can use numerical methods like the Newton-Raphson method or graphing calculators to approximate the roots.

5. Interpretation of the Roots

  • The solutions (roots) of the equation represent the x-values where the original equation is true.

Important Note: Solving this equation might lead to multiple roots, some of which could be real or complex.

In summary:

  • Simplify the equation by expanding the squares.
  • Expand the entire polynomial.
  • Convert to a standard form.
  • Use factoring, the rational root theorem, or numerical methods to find the roots.
  • Interpret the roots in the context of the original equation.

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