f(x)=x^4 -x^3 -6x^2 +4x +8 rational Zeros Theorem

Question

using the rational zeros theorem to find all real zeros of the polynomial function. use zeros to factor f.   f(x)= x^4 - x^3- 6x^3+4x+8

Answer 1

Assume you mean f(x)=x⁴-x³-6x²+4x+8.

Define set P as all the factors of the constant so P={ 1, 2, 4, 8 }.

Define set Q as all the factors of the coefficient of x⁴ so Q={ 1 }.

Create all possible ±p/q where p is in P and q is in Q:

1, 2, 4, 8, -1, -2, -4, -8.

Use synthetic division to decide which, if any, are actual factors:

-2 | 1 -1 -6 4  8

      1 -2  6 0 -8

      1 -3  0 4 

-1 | 1 -1 -6 4  8

      1 -1  2 4 -8

      1 -2 -4 8 

2 | 1 -1 -6  4  8

     1  2  2 -8 -8

     1  1 -4 -4

Only those divisions have been shown which confirm an actual zero (to save space and time in the answer). The rational zeroes are -2, -1, 2 (duplicate). (x-2)² is a factor because:

2 | 1 1 -4 -4

     1 2  6  4

     1 3  2       2 repeats as a zero.