The method is easy, but the format is difficult to show. I'll do my best.
(x^2 -3x -18) div (x-6)(x2−3x−18)÷(x−6)
" (dividend) " div " (divisor)" (dividend) ÷ (divisor)
color(magenta)("step 1:")step 1: Dividend must be in descending powers of x.
color(white)(xxxxxxxxxxxxxxxxxxxxxxxx)x^2 " "-3x" " -18××××××××××××x2 −3x −18
Only use the numerical coefficients rArr 1" "-3" "-18 "⇒1 −3 −18
(If there are any missing, leave a space or fill in a zero).
color(orange)("Step 2")Step 2: Make the divisor = 0. " "(x-6) = 0 rArr x = color(orange)(6) " this goes outside" (x−6)=0⇒x=6 this goes outside
color(white)(xxxxxxxxx) | color(brown)(1)" "-3" "-18 " "color(magenta)("step 1")××××x∣1 −3 −18 step 1
color(white)(xxxxxx)color(orange)(6) " "| darr " "color(red)(6) " "color(blue)(18)×××6 ∣⏐↓ 6 18
color(white)(xxxxxxxxxx) ul(" ")
color(white)(xxxxxxxxxxx) color(brown)(1) " "color(blue)(3) " "color(teal)(0) larr "no remainder!"
color(white)(xxxx.. xxxxx)uarr " "uarr
color(white)(xxxxxxxxxxx) x " "x^0
Step 3: Begin the division:
-"Bring down the " color(brown)( 1
) " to below the line"
-"multiply " color(orange)(6) xx color(brown)(1) = color(red)(6)
"Add" -3+color(red)(6) = color(blue)(3)
-"multiply " color(orange)(6) xx color(blue)(3) = color(blue)(18)
"Add" -18+color(blue)(18) = color(teal)(0)
That's it Folks!
We have now found the numerical coefficients of the terms in the quotient (answer)
We divided an expression with x^2 by an expression with x,
so the first term will be x^2/x = x
The last value is the remainder. In this case it is color(teal)(0)
This means that x-6 is a factor of x^2 -3x -18
(x^2 -3x -18) div (x-6) = color(brown)(1)x +color(blue)(3), "rem " color(teal)(0)
