Using the Fundamental Theorem of Algebra. Complete the following exercises. Show your work. Determine how many, what type, and find the roots for f(x) = x^4 + 21x^2 − 100.

Determine how many, what type, and find the roots for f(x) = x^3 − 5x^2 − 25x + 125.

A) - Quartic polynomial

- has four roots

- roots are; f(x) = (x + 5i)(x - 5i)(x + 2)(x - 2)

B) - Cubic polynomial

- has 3 roots

- roots are; f(x) = (x - 5)²(x + 5)

Step-by-step explanation:

A) f(x) = x⁴ + 21x² − 100

The highest power in this polynomial is 4 and thus, it is called a quartic polynomial. This means that it will have 4 roots.

Factorizing this polynomial gives;

f(x) = (x² + 25)(x² - 4)

Equating to zero, we can find the roots. Thus, the roots are;

x = 5i

x = -5i

x = 2

x = -2

Thus, the completely factorized polynomial is;

f(x) = (x + 5i)(x - 5i)(x + 2)(x - 2)

B) f(x) = x³ − 5x² − 25x + 125

The highest power in this polynomial is 3. It is therefore a cubic polynomial with 3 roots.

Factorizing this polynomial gives;

f(x) = (x² - 25)(x - 5)

Equating to zero and Factorizing gives;

x = 5

x = -5

x = 5

Thus, completely factorized with the roots is;

f(x) = (x - 5)²(x + 5)