How do you divide #(2x ^ { 3} + 4x ^ { 2} - x + 4) \div ( x + 2)#?

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Answer 1 · 2016-12-06T09:15:12

The remainder is #=6# and the quotient is #=2x^2-1#

You can use the remainder theorem, if you have a polynomial

#f(x)# and you divide by #(x-a)#, then

#f(x)=(x-a)q(x)+r#

#q(x)# is the quotient

and #r# the remainder

We have, #f(c)=r#

Here, #f(x)=2x^3+4x^2-x+4#

If we divide by #(x+2)#

#f(-2)=2*-8+4*4+2+4=-16+16+6=6#

The remainder is #=6#

We can do a long division to find the quotient

#color(white)(aaaa)##2x^3+4x^2-x+4##∣##x+2#

#color(white)(aaaa)##2x^3+4x^2##color(white)(aaaaaaa)##∣##2x^2-1#

#color(white)(aaaaaaa)##0+0-x+4#

#color(white)(aaaaaaaaaaaaa)##-x-2#

#color(white)(aaaaaaaaaaaaaaa)##0+6#

The remainder is #=6# and the quotient is #=2x^2-1#

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