What is the quotient #( x ^ { 3} + 3x ^ { 2} + 5x + 3) \div ( x + 1) #?

2 Answers
May 14, 2017

The quotient is #=x^2+2x+3#

Explanation:

Let's perform a long division

#color(white)(aaaa)##x+1##color(white)(aaaa)##|##x^3+3x^2+5x+3##color(white)(aaaa)##|##x^2+2x+3#

#color(white)(aaaaaaaaa)##color(white)(aaaaaa)##x^3+x^2#

#color(white)(aaaaaaaaa)##color(white)(aaaaaaa)##0+2x^2+5x#

#color(white)(aaaaaaaaa)##color(white)(aaaaaaaaa)##+2x^2+2x#

#color(white)(aaaaaaaaa)##color(white)(aaaaaaaaaaaa)##0+3x+3#

#color(white)(aaaaaaaaa)##color(white)(aaaaaaaaaaaaaa)##+3x+3#

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

The quotient is #=x^2+2x+3#

May 14, 2017

#x^2+2x+3#

Explanation:

One way is to use the divisor as a factor in the numerator.

#"consider the numerator"#

#color(red)(x^2)(x+1)color(magenta)(-x^2)+3x^2+5x+3#

#=color(red)(x^2)(x+1)+color(red)(2x)(x+1)color(magenta)(-2x)+5x+3#

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

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

#"quotient " =color(red)(x^2+2x+3)#