After determining the rational roots, solve the equation: 7x^5 - 27x^4 + 17x^3 + 31x^2 - 24x - 4 = 0

Question

The equation has at least two rational roots. After determining them, solve the equation.

Answer 1

f(0)=-4

f(1)=0

f(-1)=-48

f(2)=224-432+136+124-48-4=0

1 and 2 are solutions

7x^5-27x^4+17x^3+31x^2-24x-4=0

divide (x-1)(x-2) into the 5 degree polynomial to get a cbib

f(-1)=0

divide x+1 into the cubic to get

7x^2-13x-2

factor

(7x+1)(x-2)

x=-1/7, 2, 1 and -1

four solutions, 2 repeats

or graph the 5th degree polynomial and see where it intersects the x axis

(-1,0),(-1/7,0), (1,0)

and tangent at (2,0)

the x coordinates are the solutions,roots, intercepts, or zeros

all are integer factors of the constant term -4 over integer factors of the coefficient of the leading term, 7x^5

-1/1,-1/7,1/1 and 2/1