Final answer:
The equation 3z⁹ - 14z = 4z⁵ has three solutions. The correct answer is C) Three solutions.
Step-by-step explanation:
To determine how many solutions the equation 3z⁹ - 14z = 4z⁵ has, we first need to bring all terms to one side to set the equation to zero, which is the standard form for solving polynomial equations:
3z⁹ - 4z⁵ - 14z = 0
We can simplify by taking a z common from all terms:
z(3z⁸ - 4z⁴ - 14) = 0
This gives us two separate equations to solve: z = 0 and 3z⁸ - 4z⁴ - 14 = 0. The first equation gives us one solution, z = 0. The second is a polynomial of degree 8, which suggests up to 8 possible solutions. To find the exact number of distinct real solutions, further analysis or numerical methods would be needed because the second equation does not factor nicely.
To find the number of solutions for the equation 3z⁹ - 14z = 4z⁵, we need to determine the highest power of the variable, which is z.
In this equation, we have a ninth-degree polynomial on the left side and a fifth-degree polynomial on the right side. The number of solutions is equal to the degree of the polynomial. So, the equation has 9 solutions, which is option C: Three solutions.