How Do You Factor X^4-2x^3+2x-1 ? | Socratic

How do you factor # x^4-2x^3+2x-1# ? Algebra

2 Answers

Narad T. Oct 28, 2017

The answer is #=(x+1)(x-1)^3#

Explanation:

We need

#a^2-b^2=(a+b)(a-b)#

#a^2-2ab+b^2=(a-b)^2#

Let #p(x)=x^4-2x^3+2x-1#

Rewrite the function by moving the last term

#p(x)=x^4-1-2x^3+2x#

#p(x)=(x^4-1)-(2x^3-2x)#

Start factorising

#p(x)=(x^2+1)(x^2-1)-2x(x^2-1)#

Continue factorising

#p(x))=(x^2-1)(x^2+1-2x)#

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

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

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

Answer link Tony B Oct 28, 2017

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

Explanation:

This is just 1 approach.

Lets assume that the plot crosses the x-axis at least once (it does).

Set #y=0=x^4-2x^3+2x-1 #

Have a really good look at what is given. Notice that if we substitute 1 for #x# we have #y=(1)^4-2(1)^3+2(1)-1 = 0#

So #(x-1)(?.......)# is part of the factorisation.

Lets use what we have got to construct the original equation.

The first term is #x^4# so we may do this:

#(x-1)(color(red)(x^3)+......) = x^4-x^3.......#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ But the 2nd term is #-2x^3# so we need an extra #x^3#. Do this:

#(x-1)(x^3color(red)(-x^2)......) = x^4-x^3color(red)(-x^3+x^2).......#

#(x-1)(x^3-x^2......) = x^4-2x^3+x^2.........# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ There is no #x^2# term in the original equation. So we need to 'get rid of it. We do this by turning it into 0 so now we need #-x^2#

#(x-1)(x^3-x^2color(red)(-x)......) = x^4-2x^3+x^2color(red)(-x^2+x.........#

#(x-1)(x^3-x^2-x......) = x^4-2x^3+x.........# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We need to have not just #x# but#+2x#. So we need another #x# from somewhere.

#(x-1)(x^3-x^2-xcolor(red)(+1)) = x^4-2x^3+xcolor(red)(+x-1)#

#color(green)((x-1)(x^3-x^2-x+1) = x^4-2x^3+2x-1)# part answer

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now we need to solve for #x^3-x^2-x+1=0#

Again notice that #x=1# is a solution

#(+1)^3-(+1)^2-(+1)+1=0#

#color(white)("dd")1color(white)("dddd")-1color(white)("ddddd")-1color(white)("d.")+1=0#

So we can go through the process again.

#(x-1)(color(red)(x^2)..... )=color(red)(x^3-x^2)....#

#(x-1)(x^2color(red)(-1 ) )=x^3-x^2 color(red)(-x+1)#

#color(green)((x-1)(x^2-1 )=x^3-x^2 -x+1)# the other part answer ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Putting it all together")#

#(x-1)^2(x^2-1)#

But this is the same as

#(x-1)^2(x^2-[1^2])#

#(x-1)^2(x+1)(x-1)#

Which is:

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

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