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

1 Answer
Apr 21, 2018

Answer:

#(2x+1)(x+1)(x-1/2-1/2sqrt3i)(x-1/2+1/2sqrt3i)#

Explanation:

#color(blue)"factor by grouping"#

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

#"take out the "color(blue)"common factor "(2x+1)#

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

#x^3+1" is a "color(blue)"sum of cubes"#

#•color(white)(x)a^3+b^3=(a+b)(a^2-ab+b^2)#

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

#"we can factor "x^2-x+1" by solving "x^2-x+1=0#

#"using the "color(blue)"quadratic formula"#

#"with "a=1,b=-1" and "c=1#

#rArrx=(1+-sqrt(1-4))/2=(1+-sqrt3i)/2=1/2+-1/2sqrt3i#

#(x-(1/2+1/2sqrt3i))(x-(1/2-1/2sqrt3i))#

#=(x-1/2-1/2sqrt3i)(x-1/2+1/2sqrt3i)#

#rArr2x^4+x^3+2x+1#

#=(2x+1)(x+1)(x-1/2-1/2sqrt3i)(x-1/2+1/2sqrt3i)#