fx 2x5 x4 22x3 18x2 48x 45 a 2 points List all possible

f(x) = 2x⁵ + x⁴ 22x³ 18x² + 48x + 45

(a) (2 points) List all possible rational zeros of this polynomial.

(b) (8 points) Find all zeros of this polynomial by using synthetic division to factor. Be sure to show all your work. Make sure your final answer includes both a list of all zeros and the completely factored form of the polynomial.

Solution

3, sqrt3 , -sqrt3 all are roots of the given polynomial

now 2x⁵ + x⁴ 22x³ 18x² + 48x + 45/(x-3)(x²-3)

=2x² + 7x + 5

so 2 more root are ,,, (-7 + sqrt(49-40))/4 , (-7 - sqrt(49-40))/4

i.e -1, -2.5