To divide (x^2 +4x + 4)/(x+2)
Write the divisor, x+2 on the left. Then the dividend, x^2+4x+4 under the division thing. (I don't know its name.)
" " " "-----
x+2 ) x^2 +4x +4
Look at the first terms on the divisor x+2 and the dividend x^2+4x+4. What do we need to multiply x by, to get x^2? We need to multiply by x. Write that x on the top line:
" " " " " " "x
" " " "-----
x+2 ) x^2 +4x +4
Now multiply x times the divisor, x+2, to get x^2+2x and write that under the dividend.
" " " " " " " x
" " " "-----
x+2 ) x^2" " +4x" " +4
" " " " "" " x^2 " " +2x
" " " "-----
Now we need to subtract x^2+2x from the dividend. (You may find it simpler to change the signs and add.)
" " " " " " " x
" " " "-----
x+2 ) x^2 " "+4x " "+4
" " " " color(red)(-)x^2color(red)(-)2x
" " " "-----
" "" "" "" "" " 2x" " +4
Now, look at the first terms of the divisor and the expression on the last line. What do we need to multiply x (the first term of the divisor) by to get 2x (the first term on the last line)? We need to multiply by 2
Write +2 on the top line, then multiply 2 times the divisor x+2, to get 2x+4 and write it underneath.
" " " " " " " x" " +2
" " " "-----
x+2 ) x^2 " "+4x " "+4
" " " " color(red)(-)x^2color(red)(-)2x
" " " "------
" " " " " "" " 2x" " +4
" " " " " " " " 2x" " +4
" " " "------
Now subtract (change the signs and add), to get:
" " " " " " " x" " +2
" " " "-----
x+2 ) x^2 " "+4x " "+4
" " " " color(red)(-)x^2color(red)(-)2x
" " " "------
" " " " " "" " 2x" " +4
" " " " " " color(red)(-)2x" " color(red)(-)4
" " " "------
" " " " " " " " " "" " " " 0
The quotient is x+2 and the remainder is 0.
