Make a list of all possible rational zeros of f(x)=5x^3+7x^2−81x+45, and then use this list to find all of the functions’s rational zeros, if any exist.

Question

Rational Zero Candidates: ±

Rational Zeros:

Answer 1

There are too many to list. You can do that.

I used 15 as a factor from 45 and 5 as a factor from 5.

15/5=3

3 is a root

5(27)+7(9)-81(3)+45=135+63-243+45

add the positive integers to get 243 and 243-243=0

therefore 3 is a root and the original equation is divisible by x-3

divide the original equation by x-3 to get 5x^2+22x-15

now factor this equation: 5x^2+22x-15

5*15=5*3*5=3*(5*5)=3*25 and 25-3=22, the coefficient of the middle term

factor: (5x )(x 5) because the 5's need to multiply each other

continuing on you get (5x-3)(x+5) because 25 has to be positive and 3 has to be negative

therefore 5x-3=0 so x=3/5 and x+5=0 so x=-5

the roots are x=3, x=-5 and x=3/5

Answer 2

desmos.com/calculator/bqvpetpgqm