Question

Determine the number of roots of each polynomial equation.

3. 5x⁴(x-7)(x²+8)²=0
A. 3
B. 4
C. 5
D. 6

Answer 1

The polynomial equation 5x⁴(x-7)(x²+8)²=0 has five real roots, with four being at x=0 and one at x=7. The correct answer is option C, which represents 5 roots.

Step-by-step explanation:

The question asks to determine the number of roots of the polynomial equation 5x⁴(x-7)(x²+8)²=0.

To find the number of roots, we look at the given factors of the polynomial:

1. The term 5x⁴ suggests that there are four roots at x=0 (also called the zero of the polynomial).
2. The linear term (x-7) indicates one root at x=7.
3. The term (x²+8)² is a perfect square, indicating that the quadratic equation has a double root. Since x²+8 cannot be zero for any real number x (as the smallest value x² can have is zero, making the smallest value of x²+8 equal to 8), it contributes no real roots.

Thus, the polynomial has a total of five real roots: four at x=0 and one at x=7. The correct answer is therefore option C (5 roots).

When solving polynomial equations, such as ax²+bx+c=0, the quadratic formula can be used to find the solutions.

However, in this case, the factors provided allow us to determine the roots directly without resorting to the formula.