How do you factor #y= 2x^3 - 5x^2 + 4 #?

2 Answers
Feb 29, 2016

Answer:

#y= (x-2)(2x^2-x-2)#

Explanation:

#y=2x^3-5x^2+4 = 2x^3-4x^2-x^2+4 = 2x^2(x-2) - (x^2-4) =2x^2(x-2) - ((x+2)(x-2)) = (x-2)(2x^2-(x+2) = (x-2)(2x^2-x-2)#[Answer]

Aug 18, 2016

Let #g(x)=y=2x^3-5x^2+4#.

We observe that sum of the co-effs.#=2-5+4=1!=0, so, (x-1)# is not a factor.

Also, the sum co-effs. of odd powered terms #=2!=-1=-5+4# = the sum of co-effs. of even powered terms, so #(x+1) can not be a factor.

Now to guess the probable factors of #g(x)#, we look at its leading co-eff. #=2,# having factors #1,2# and the const. term#=4,# having factors #1,2,4.#

The probable factors can be guessed by multiplying #x# with the factors of leading co-eff. and taking with this, #+-# the factors of the const. term. This leads us to the guess of probable factors of #g(x)# as #x+-1,x+-2,x+-4, 2x+-1,2x+-2,2x+-4.#

Among these, we have already verified the factors #x+-1, 2x+-2=2(x+-1).#

Next we verify whether #(x-2)# is a factor of #g(x)# by checking #g(2)=2*8-5*4+4=16-20+4=0,#, so, (x-2) is a factor.

Now we arrange the terms of #g(x)# in such a way (shown below) that #(x-2)# turns out as a common factor :-

#g(x)=2x^3 -5x^2+4=ul(2x^3-4x^2)-ul(x^2+2x)-ul(2x+4)=2x^2(x-2)-x(x-2)-2(x-2)=(x-2)(2x^2-x-2)#.

Enjoy maths.!