How do you factor #x^3 – 4x^2 +x + 6#?

2 Answers
Mar 29, 2017

Answer:

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

Explanation:

It is usually really, really hard to factorize a cubic function. However, for this polynomial, we can factor by grouping. We try values for splitting the term #-4x^2#.

For example, we split it into #-2x^2-2x^2#.

The equation becomes this: #(x^3-2x^2)-(2x^2-x-6)#. We can factorize each of the expressions in the parentheses: #x^2(x-2)-(x-2)(2x+3)#. There is a common factor #(x-2)#.

Factoring the common factor out, we get #(x-2)(x^2-2x-3)#. We then factorize #x^2-2x-3# to #(x-3)(x+1)#.

The fully factored form is then #(x-2)(x-3)(x+1)#.

Mar 29, 2017

Answer:

#x^3-4x^2+x+6=color(magenta)((x-2)(x-3)(x+1))#

Explanation:

Provided the expression has rational roots, we can use the rational root theorem.

For the expression #color(green)(x^3-4x^2+x+6)#
according to the rational root theorem, possible rational roots are:
#color(white)("XXX"){+-1,+-2,+-3,+-6}#

With the use of a spreadsheet these values can be easily checked (it can also be done with a calculator or even manually with a bit more effort).
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From this, we see that #x=2#, #x=3#, and #x=-1# all are zeros for the given expression.
This implies that #(x-2)#, #(x-3)#, and #(x+1)# are factors of the given expression.

Since #x^3-4x^2+x+6# is of degree #3#, it has a maximum of #3# factors and we have found them all.