To evaluate f(x)=-x^5-4x^3+6x^2-x at x=-2 using direct substitution, we just put value of x in the polynomial.
Hence, f(-2)=-(-2)^5-4(-2)^3+6(-2)^2-(-2)
= -(-32)-4(-8)+6xx4+2
= 32+32+24+2=90
Before we talk about synthetic division, one thing regarding remainder theorem which states that when a polynomial f(x) is divided by x-a, the remainder is f(a).
Hence, to find f(-2), we should divide f(x) by x-(-2)=x+2, the remainder so obtained is f(-2). Hence let us divide f(x)=-x^5-4x^3+6x^2-x by x+2 using synthetic division method.
One Write the coefficients of x in the dividend inside an upside-down division symbol. We are using 0 as coefficient of x^4 and constant term.
color(white)(1)|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0
color(white)(1)|" "color(white)(X)
" "stackrel("——————————————————------)
Two As x+2=0 gives x=-2 put -2 at the left.
-2|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0
color(white)(xx)|" "color(white)(XX)
" "stackrel("—————————————---------)
Three Drop the first coefficient of the dividend below the division symbol.
-2|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0
color(white)(xx)|" "color(white)(X)
" "stackrel("—————————————---------)
color(white)(xx)|color(white)(xX)color(red)(-1)
Four Multiply the result by the constant, and put the product in the next column.
-2|color(white)(X)-1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0
color(white)(xx)|" "color(white)(xxXxx)2
" "stackrel("————————————---------—)
color(white)(xx)|color(white)(X)color(blue)-1
Five Add down the column.
-2|color(white)(X)1" "color(white)(X)0color(white)(XX)-4" "" "6" "" "-1" "" "0
color(white)(xx)|" "color(white)(xXx)2
" "stackrel("—————————————---------)
color(white)(xx)|color(white)()color(blue)-1color(white)(X1)color(red)2
Six Repeat Steps Four and Five until you can go no farther.
-2|color(white)(X)1" "color(white)(X)0color(white)(X)-4" "" "6" "" "-1" "" "0
color(white)(xx)|" "color(white)(Xxx)2color(white)(xx)-4color(white)(xxx)16color(white)(xxx)-44color(white)(xxx)90
" "stackrel("———--------——————————)
color(white)(xx)|color(white)()color(blue)-1color(white)(xxx)color(red)2color(white)(X)color(red)-8color(white)(XX)color(red)22color(white)(XX)color(red)-45color(white)(xxx)90
Hence, Quotient is -x^4+2x^3-8x^2+22x-45 and remainder is 90 an hence f(-2)=90