What are all the possible rational zeros for #f(x)=3x^3+11x^2+5x-3# and how do you find all zeros?

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Answer 1 · 2016-09-07T14:21:54

The Zeroes of #f# are #1/3,-3,&,-1#.

We will factorise #f(x)=3x^3+11x^2+5x-3,# to find its zeroes.

Observe that,

The Sum of the co-effs. of Odd-powered terms#=3+5=8,# and,

that of the Even-powered ones#=11-3=8#.

Hence, #(x+1)# is a factor of #f(x)#.

Now, #f(x)=3x^3+11x^2+5x-3#

#=ul(3x^3+3x^2)+ul(8x^2+8x)-ul(3x-3)#

#=3x^2(x+1)+8x(x+1)-3(x+1)#

#=(x+1)(3x^2+8x-3)#

#=(x+1){ul(3x^2+9x)-ul(x-3)}#

#=(x+1){3x(x+3)-1(x+3)}#

#=(x+1)(x+3)(3x-1)#

Hence, the Zeroes of #f# are #1/3,-3,&,-1#.