Use this function to answer questions a - d. Please show your work
f(x) = 3x^3 - 7x^2 - 14x + 24
a. Using the Fundamental Theorem of Algebra. How many possible complex roots will this polynomial have?
b. Using Descartes Rule, determine how many possible positive roots and how many possible negative roots this function will
have.
c. Using the Rational Roots Theorem, list all the possible rational roots for the function:
d. Using the possible roots you found on the previous question, find all the possible zeros for this function. (Hint: use synthetic division)
a. According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n complex roots. Since the degree of the polynomial f(x) is 3, it will have 3 possible complex roots.
b. To determine the possible number of positive and negative roots using Descartes Rule, we need to examine the signs of the coefficients of the polynomial.
For f(x) = 3x^3 - 7x^2 - 14x + 24:
- The signs of the coefficients in f(x) remain the same when we change the sign of x (i.e., f(-x)).
- The number of sign changes in f(x) is 2.
Therefore, using Descartes Rule, this polynomial has 2 possible positive roots (or less), and 2 possible negative roots (or less).
c. The Rational Roots Theorem states that the possible rational roots of a polynomial with integer coefficients are all the possible ratios of the factors of the constant term (in this case, 24) to the factors of the leading coefficient (in this case, 3).
Factors of 24: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
Factors of 3: ±1, ±3
Possible rational roots: ±1/1, ±1/3, ±2/1, ±2/3, ±3/1, ±3/3, ±4/1, ±4/3, ±6/1, ±6/3, ±8/1, ±8/3, ±12/1, ±12/3, ±24/1, ±24/3
d. To find the possible zeros using synthetic division, we will divide the polynomial f(x) by each of the possible rational roots obtained in the previous question.
For example, let's try dividing f(x) by x = 2/3 using synthetic division:
2/3 | 3 -7 -14 24
--------------------
3 -1 -16 10
The result of the synthetic division is 3x^2 - x - 16 with a remainder of 10.
We repeat this process for each of the possible rational roots to find the possible zeros of the function.