What is the quotient #(3x^4-4x^2+ 8x-1)div (x-2)#?
1 Answer
Explanation:
We set up the long division of a polynomial by a simple monomial like this:
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
#color(white)(SPACE)3x^3#
#(x-2))bar(3x^4-4x^2+8x-1)#
Now, we multiply
#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
Now, we check: how many times does our leading coefficient
#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
#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
#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
#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
#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
#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,
The final answer is therefore