Remember to write any terms whose coefficient is 0:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) color(white)( (x^4-5x^3+6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
Because #x^2*x^2 = x^4#, the first term of the quotient is #x^2#:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2color(white)( 5x^3+6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
Multiply the first term in the quotient by the divisor and subtract underneath:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2color(white)( 5x^3+6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
#color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr#
#color(white)(.........................)-5x^3+13x^2-10x#
Because #-5x*x^2 =-5x^3#. the next term in the quotient is #-5x#:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5xcolor(white)(6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
#color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr#
#color(white)(.........................)-5x^3+13x^2-10x#
Multiply the second term in the quotient by the divisor and subtract underneath:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5xcolor(white)(6x^2-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
#color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr#
#color(white)(.........................)-5x^3+13x^2-10x#
#color(white)(..............................)ul(5x^3+color(white)(.)0x^2-45x)" "darr#
#color(white)(........................................)13x^2-45x+15#
Because #13*x^2 =13x^2#. the next term in the quotient is #13#:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5x+13color(white)(-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
#color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr#
#color(white)(.........................)-5x^3+13x^2-10x#
#color(white)(..............................)ul(5x^3+color(white)(.)0x^2-45x)" "darr#
#color(white)(........................................)13x^2-45x+15#
Multiply the third term in the quotient by the divisor and subtract underneath:
#color(white)( (x^2+0x-7)/color(black)(x^2+0x-7)) (x^2-5x+13color(white)(-10x+15))/( ")" color(white)(")")x^4-5x^3+6x^2-10x+15)#
#color(white)(...................)ul(-x^4-0x^3+7x^2)" "darr#
#color(white)(.........................)-5x^3+13x^2-10x#
#color(white)(..............................)ul(5x^3+color(white)(.)0x^2-35x)" "darr#
#color(white)(........................................)13x^2-45x+15#
#color(white)(.....................................)ul(-13x^2color(white)(.)-0x+91)#
#color(white)(...............................................)-45x+106#
The quotient is #x^2-5x+13# with a remainder of #-45x+10#
