Question

Find the bounds on the real zeros of the following function. Please show all of your work. f(x)=x^5+3x^4-3x+2

Answer 1

If you have a polynomial with real coefficients and a positive leading coefficient, then ...

Upper Bound

If synthetic division is performed by dividing by x-k, where k>0, and all the signs in the bottom row of the synthetic division are non-negative, then x=k is an upper bound (nothing is larger) for the zeros of the polynomial.

Lower Bound

If synthetic division is performed by dividing by x-k, where k<0, and the signs in the bottom row of the synthetic division alternate (between non-negative and non-positive), then x=k is a lower bound (nothing is smaller) for the zeros of the polynomial.

The zero in the bottom row may be considered positive or negative as needed.

http://people.richland.edu/james/lecture/m116/polynomials/zeros.html

Answer 2

Please help with this: List the potential rational zeros of the following function. Please explain. f(x)=2x^5-6x^2+2x-1

Answer 3

Rational Root Test

If a polynomial function has integer coefficients, then every rational zero will have the form p/q where p is a factor of the constant and q is a factor of the leading coefficient.

•Make sure the polynomial has integer coefficients. Multiply to get rid of fractions or decimals if need be (be sure to later divide).

•This only addresses the rational zeros.

• What it does give you is a list of possible rational zeros

Answer 4

Factors of 1: 1 and -1

Factors of 2: 1 and -1 and 2 and -2

Possible Rational Roots:

1/1, 1/(-1), 1/2, 1/(-2)

-1/1, -1/-1, -1/1, -1/-1. 1/2, 1/(-2), -1/2, -1/(-2)

Clearing out the repeated candidates:

1, -1, 1/2, -1/2 are the possible rational roots

Answer 5

Check yourself by doing the following:

List the possible rational roots for y = 4x^2 + x + 3

Begin by getting the factors of 3.

Then, get the factors of 4.

Then, make ratios (fractions) of the factors of 3 over the factors of 4.

Note: It's messy but be systematic and you won't miss any.