Question

The equation \(2x^5 + 5x = 3x^3 + 4x^4\) has:

• A. no real solution
• B. only one non-zero real solution
• C. infinitely many solutions
• D. only three non-negative real solutions

Hint:

Factorization and numerical methods are helpful in solving higher-degree polynomial equations.

Answer 1

The Correct Option is B

Step 1: The initial equation is \(2x^5 + 5x = 3x^3 + 4x^4\). To begin solving, rearrange the terms:

\[ 2x^5 + 5x - 3x^3 - 4x^4 = 0 \]

Step 2: Factor the equation:

\[ x(2x^4 + 5 - 3x^2 - 4x^3) = 0 \]

This yields one solution: \(x = 0\).

Step 3: The remaining equation is \(2x^4 - 4x^3 - 3x^2 + 5 = 0\). Numerical methods or graphing indicates only one non-zero real solution exists.

Step 4: Consequently, the original equation possesses only one non-zero real solution.