What is the quotient #(3x^4-4x^2+ 8x-1)div (x-2)#?

1 Answer
Dec 26, 2016

#3x^3+6x^2+8x+24 + (47)/(x-2)#

Explanation:

We set up the long division of a polynomial by a simple monomial like this:

#(x-2))bar(3x^4-4x^2+8x-1)#

It works just like the long (numerical) division most of us learned back in elementary school, except now we're dividing with variables.

First we check: how many times does our leading #x# term in the divisor, in this case just #x# (coefficient of #1#), go into our leading #x# term in the dividend, #3x^4#? We would have to multiply #x# by #3x^3# to get #3x^4#. Just like in long division with integers, we put this above the bar.

#color(white)(SPACE)3x^3#
#(x-2))bar(3x^4-4x^2+8x-1)#

Now, we multiply #3x^3# by the divisor and subtract that from #3x^4#.

#color(white)(SPACE)3x^3#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^3(x-2)#

#=>#

#color(white)(SPACE)3x^3#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#

We can subtract #3x^4# from #3x^4# to get #0#. We see that we have no #x^3# term in the dividend, so we treat this as though we're adding #6x^3# to #0#.

Now, we check: how many times does our leading coefficient #x# in the divisor go into #6x^3#? We would have to multiply #x# by #6x^2# to get #6x^3#. Similarly to above:

#=>#

#color(white)(SPACE)3x^3+6x^2#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACE)-6x^2(x-2)#

#=>#

#color(white)(SPACE)3x^3+6x^2#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACES)-6x^3+12x^2#

This gets rid of our #x^3# term, but now we have an #x^2# term. We also have an #x^2# term in the dividend, so we can add this to the current remainder.

#=>#

#color(white)(SPACE)3x^3+6x^2#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPA)+12x^2#

#=>#

#color(white)(SPACE)3x^3+6x^2#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#

Next we check: how many times does our leading coefficient #x# in the divisor go into #8x^2#? We would have to multiply #x# by #8x# to get #8x^2#.

#=>#

#color(white)(SPACE)3x^3+6x^2+8x#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESP)-8x(x-2)#

#=>#

#color(white)(SPACE)3x^3+6x^2+8x#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESP)-8x^2+16x#

#=>#

#color(white)(SPACE)3x^3+6x^2+8x#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESPACES)+16x#

Now we add #16x# to the #8x# term in the dividend.

#=>#

#color(white)(SPACE)3x^3+6x^2+8x#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESPACESPA)24x#

Next we check: how many times does our leading coefficient #x# in the divisor go into #24x#? We would have to multiply #x# by #24# to get #24x#.

#=>#

#color(white)(SPACE)3x^3+6x^2+8x+24#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESPACESPA)24x#
#color(white)(SPACESPACE)-24(x-2)#

#=>#

#color(white)(SPACE)3x^3+6x^2+8x+24#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESPACESPA)24x#
#color(white)(SPACESPACE)-24x+48#

#=>#

#color(white)(SPACE)3x^3+6x^2+8x+24#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESPACESPA)24x#
#color(white)(SPACESPACESPACE)48#

Now we add #48# to the #-1# in our dividend.

#=>#

#color(white)(SPACE)3x^3+6x^2+8x+24#
#(x-2))bar(3x^4-4x^2+8x-1)#
#color(white)(SPA)-3x^4+6x^3#
#color(white)(SPACESPA)6x^3#
#color(white)(SPACESPAC)8x^2#
#color(white)(SPACESPACESPA)24x#
#color(white)(SPACESPACESPACE)47#

And of course, #x# goes into #47# zero times. This leaves us with a remainder.

The final answer is therefore #3x^3+6x^2+8x+24 + (47)/(x-2)#