College Alebra Question PLEASE SHOW ALL WORK do NOT skip or

College Alebra Question. PLEASE SHOW ALL WORK (do NOT skip or combine any steps). I know the answer but I don\'t know how they arrived at that answer.

Again, I need to see ALL of the steps for how to arrive at this answer.

16. The polynomial 4x 4x 19x 14x 3 has four rational zeros. Find the zero that has multiplicity two. ANSWER 1/2

Solution

4x^4+4x^3-19x^2+14x-3

the possible rational zeros of the polynomial are

+- { 1,3} / { 1,2,4}

+- { 1, 1/2 ,1/4 , 3 , 3/2 ,3/4}

the polynomial has real zero at x = 1

to find other zeros divide the polynomial by x-1

4x^4+4x^3-19x^2+14x-3 / ( x-1) = 4x^3+8x^2-11x +3

again finding the rational zeros of 4x^3+8x^2-11x +3

+- { 1,1/2,1/4,3,3/2,3/4}

real zero is at x = 1/2

dividing the polynomial 4x^3+8x^2-11x +3 by 2x-1

we get 2x^2+5x-3

solving the quadratic 2x^2+5x-3 for x

2x^2+6x-x-3

2x(x+3)-1(x+3)

(2x-1)(x+3)

hence other two zeros are

x = 1/2

x = -3

so 4 zeros of the polynomial are

x = 1

x = 1/2

x = 1/2

x = -3

hence , x = 1/2 has multiplicity 2