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 given equation is \(2x^5 + 5x = 3x^3 + 4x^4\). To solve it, first move all terms to one side:

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

Step 2: Factor the equation:

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

This gives one solution \(x = 0\).

Step 3: For the remaining equation \(2x^4 - 4x^3 - 3x^2 + 5 = 0\), numerically solving it or using graphing tools, we find that the equation has only one non-zero real solution.

Step 4: Therefore, the equation has only one non-zero real solution.

Answer 2