For the polynomial function, (a) list all possible rational zeros, (b) find all rational zero, and (c) factor f(x)! f(x)=6x^3+13x^2_41x+12

numerator must be 1,2,3,4,6,12

denominator must be in 1,2,3,6
+ or - also possible

look for the easy ones first. I usually eschew fractions, so some trial and error produced

6x^3+13x^2-41x+12 = (x+4)(6x^2-11x+3)

Now you can solve the quadratic by factoring or formula.

To find the possible rational zeros for a polynomial function, you need to consider the factors of the constant term divided by the factors of the leading coefficient.

(a) List all possible rational zeros:
The constant term of the given polynomial is 12, and the leading coefficient is 6. So, you need to find the factors of 12 and divide them by the factors of 6.

Factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
Factors of 6: ±1, ±2, ±3, ±6

Dividing the factors of 12 by the factors of 6, we get the possible rational zeros:
±1, ±2, ±3, ±4, ±6, ±12

(b) Find all rational zeros:
To find all the rational zeros of the polynomial, you can use synthetic division or any other appropriate method.

Using synthetic division or long division, you can divide the given polynomial by each of the possible rational zeros. Testing each one will help you find the actual rational zeros.

For example, let's test x = 1 as a possible rational zero using synthetic division:

1 | 6 13 -41 12
|___ 6 19 -22
6 19 -22 -10

Since the remainder is not zero, x = 1 is not a rational zero.

You can repeat this process for all the possible rational zeros until you find the actual rational zeros of the polynomial.

(c) Factor f(x):
Once you have determined the rational zeros, you can factor f(x) using the rational zeros you found.

In this case, you need to find the rational zeros first before factoring the polynomial. Once you have the rational zeros, you can apply the factor theorem or synthetic division to factor the polynomial completely.