Question

Use the fundamental Theorem of Algebra to determine the number of zeros. Then find the zeros.
f(x)= (x^2 - 2x - 24)(x^2 + 4)

Answer 1

The Fundamental Theorem of Algebra says that a polynomial has a number of roots equal to the degree of the polynomial.
To determine the degree of the polynomial, let's multiply the two quantities to get \[x^4 - 2x^3 - 20x^2 - 8x - 96\]What is the degree of this polynomial? (The degree is the value of the highest exponent.)

Answer 2

The degree is 4, which means there are 4 zeros? and then how do you find the zeros from there?

Answer 3

Correct!
We need to set the function equal to zero and solve for x \[x^4 - 2x^3 - 20x^2 - 8x - 96 = 0\]
To make things easier, lets return to our original state. Now we have two 2nd order equations to solve. We get \[x^2 - 2x - 24 = 0 ~~\text{and}~~ x^2 + 4 = 0\]

Answer 4

(x-6)(x+4)=0 and (x+2)(x-2)=0
so the zeros are 6, -4, 2, and -2?

Answer 5

Correct. Note that only imaginary roots are present in the x^2 + 4 equation.

Answer 6

Wait, why are there only imaginary roots present in x^2 + 4?

Answer 7

x^2 = -4. You can't take the square root of a negative number.

Answer 8

Can you not factor the equation so that it's (x+2)(x-2) and get the answer of +-2? If not, how come?

Answer 9

You can. Your answers are correct. I was just noting the fact that they were imaginary.