SOLUTION: Factor f(x)=3x^4-54x^2+96x-45, given that 1 is a zero of f with multiplicity 2. (Use either polynomial long division or synthetic division.)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor f(x)=3x^4-54x^2+96x-45, given that 1 is a zero of f with multiplicity 2. (Use either polynomial long division or synthetic division.)      Log On


   

Question 925084: Factor f(x)=3x^4-54x^2+96x-45, given that 1 is a zero of f with multiplicity 2. (Use either polynomial long division or synthetic division.)
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-1%29%5E2=x%5E2-2x%2B1, the square because multiplicity 2.
Perform the division, polynomial division. You must include the term for x^3 in the dividend.

_________________|________3x^2__+6x____-45__________________Quotient Line
______________________________________________
x^2-2x+1_________|________3x^4+0*x^3-54x^2+96x-45
__________________________3x^4-6x^3+ 3x^2
_____________________________0+6x^3-57x^2+96x
_______________________________6x^3-12x^2+6x
________________________________0__-45x^2+90x-45
___________________________________-45x^2+90x-45
_________________________________________________
__________________________________________0+0+0_________Remainder 0


The quotient has a scalar factor, so although factoring not yet complete, you know you have %28x-1%29%5E2%2A3%2A%28x%5E2%2B2x-15%29; which you can finish into
highlight%283%28x-1%29%5E2%28x-3%29%28x%2B5%29%29.