How do you factorise #2x^3-5x+x+2#?

2 Answers

I find it best to look at the polynomial is the following form:
#2x^3+0x^2-4x+2#

Explanation:

The best way is by long division. However, finding a factor of this polynomial is difficult.

...

After some trial and error, you will realise that #x-1# is a factor.
Therefore mathematically:
#2x^3+0x^2-4x+2#
#=(2x^3-2x^2)+(2x^2-2x)-(2x+2)#
#=(x-1)(2x^2+2x-2)#
#=2(x-1)(x^2+x-1)#

I hope this helps!

May 5, 2018

Demonstrating long division

Explanation:

Given that it is determined that #x=1# is a solution (by observation)

Giving #(x-1)("something")#

Using place keepers of 0 value. Example #0x^2#

Then we have:

#color(white)("ddddddddd.dd")2x^3+0x^2-4x+2#
#color(magenta)(2x^2)(x-1)-> ul(2x^3-2x^2color(white)("ddd")larr" Subtract")#
#color(white)("ddddddddddddd")0+2x^2-4x+2#
#color(magenta)(2x)(x-1) -> color(white)("dddd.d")ul(2x^2-2x larr" Subtrtact")#
#color(white)("dddddddddddddddddd")0-2x+2#
#color(magenta)(-2)(x-1)-> color(white)("ddddddd.d")ul(-2x+2)#

Putting it all together:

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

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

If you wished to take the quadratic 'down' further you would use the formula.

Tony B