Find all rational zeros of the polynomial. P(x) = 4x^4 − 13x^3 − 13x^2 + 52x − 12

Question

Really stuck on this one:

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = 4x⁴ − 13x³ − 13x² + 52x − 12

x = ____________

Write the polynomial in factored form.

P(x)= ______________

Thank you for your help.

Answer 1

I was lazy. I calculated function values at the integers -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12, chosen as they are factors of 12, and determined roots at -2, 2, 3. (Actually, if you use Excel, it is just as easy to check all integers from -12 to 12) Then, since the product of the roots is -12/4 = -3, -2 * 2 * 3 * x = -3. -12x = -3.

x = 1/4.

Try 1/4 to see if it is a root.

It is.

x = -2, 1/4, 2, 3

The polynomial factors as (x+2)(4x-1)(x-2)(x-3)

An alternate method to the solution.

Plug one of these values into synthetic division and get the third degree polynomial. Plug another into the third degree polynomial and get the second degree polynomial. Then, you should be able to factor the quadratic. (Factoring is made easy because you already know the solution).