Use the Rational Zero Theorem to list possible rational zeros for the polynomial function. (Enter your answers as a comma-separated list.)

Question

P(x)+3x³+25x²-3x-25

Hint- You need to list all the values p/q. p is the factors of 25 and q is the factors of 3. You need all combinations here. Make sure to have all positive and negative of each one.

Answer 1

With a constant of -25 and a leading coefficient of 3, the possible rational roots are

1, 5, 25, 1/3, 5/3, 25/3, -1, -5, -25, -1/3, -5/3, -25/3

If we want to show these in increasing order, the possible rational roots are

-25, -25/3, -5, -5/3, -1, -1/3, 1/3, 1, 5/3, 5, 25/3, 25

The pattern 3 25 -3 -25 is pretty revealing.

Factoring the first two, then looking at the next two terms, we get

x^2(3x + 25) - 1(3x + 25) =

(x^2 - 1)(3x + 25) =

(x - 1)(x+1)(3x + 25)

Zeroes are -1, 1, -25/3