Find all the zeros of the function f(x)=x^4-5x^3+3x^2+19x-30 if x = -2 and x = 3 are zeros

Question

guys this is the last one and i cant solve it help

Answer 1

Here's a hint that might help: You'll need some synthetic division.

You've been told that -2 and 3 are roots. Use these as the divisors for synthetic division of the original polynomial.

-2 _| 1 -5 3 19 -30

-2 14 -34 30

------------------------

1 -7 17 -15 0

Then take that result and do synthetic division by 3 (the final 0 is omitted because that means no remainder term in the previous division; -15 is the constant term of this new polynomial):

3 _| 1 -7 17 -15

3 -12 15

--------------------

1 -4 5 0

No remainder! Fantastic. So here's what this whole thing means. When I take the original polynomial and determine that -2 and 3 are roots, that means that (x + 2) and (x - 3) are roots, so the other factor in this picture is (x² - 4x + 5).

So far, our factoring shows this: (x + 2)(x - 3)(x² - 4x + 5).

Determine if x² - 4x + 5 is factorable. It isn't; take it to the quadratic formula (let a = 1, b = -4, and c = 5).

Unfortunately, the discriminant (the b² - 4ac) is -4, and the square root of this is 2i. You have two complex roots on your hands! Simplify the results, noting what the 4, 2i, and 2 all have a common factor of 2:

x = (4 ± 2i)/2

x = 2(2 ± i)/2

x = 2 ± i

Our roots are x = -2, 3, 2 + i, 2 - i.