Use the following polynomial equation to do the following

Question

Use the following polynomial f(x)=5x^3+8x^2-4x+3

a. the fundemental theorem of algebra states that this polynomial has ____ roots.

b. find f(-x).

c. use descartes' rule of signs to complete the table:
number of positive real zeros, number of negative real zeros, number of imaginary zeros
1
2
3

d. use the rational root theorem to determine the possible rational roots of f(x).

e. of the possible rational roots above, which ones are roots?

Answer 1

Given  f(x)=5x^3+8x^2-4x+3 

(a).The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree  and hence the given polynomial has 3 roots

So there is only 1 sign change in it so therfore there will always be one NEGATIVE root because you can't subract 2 from it because then there would be -1 negative roots and that's not possible. From there you can find the imaginary roots: 

If there are 2 positive and 1 negative there are 0 imaginary 

If there are 0 positive and 1 negative there are 2 imaginary 
 

 

Answer 2

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

d. 5x³+8x²-4x+3

If p/q is a rational zero, then p is a factor of 3 and q is a factor of 5.

The possible values of p are ± 1, ±3. The possible values for q are ± 1, ± 5.

So, possible rational roots are, p/q = ±1, ±3, ±1/5, ±3/5.

Comments

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

Answer 3

e.

Make a table for the synthetic division and test possible zeros.

Non of the possible rational roots giving zero remainder, so these are not the roots of the polynomial.