How do you divide #(x^2 - 2x - 15)/(x + 3)# using polynomial long division?

Question

How do you divide #(x^2 - 2x - 15)/(x + 3)#  using polynomial long division?

Eva Abramson · Accepted Answer

Answer 2 of 2

#x-5#

Have a look at the method. It shows a useful 'trick'.

Given: #(x^2-2x-15)/(x+3)........................(1)# Not all questions permit this approach of solution! Consider #color(white)(..)x^2-2x-15# This can be factored into: #(x-5)(x+3).............................(2)# Substitute expression (2) into expression (1) #((x-5)(x+3))/(x+3)# Write as: #(x+3)/(x+3) xx (x-5)# But #(x+3)/(x+3)# has the value of 1 giving #1xx (x-5)# #x-5# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Foot note")# Consider: #(x+3)/(x+3)# If you were investigating values then this produces a problem. You are not mathematically allowed to divide by 0. So #(x+3)/(x+3)# is 'Undefined' at #x=-3# For this very reason #0/0# does #underline(color(red)("not equal 1"))# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Ethan Adkins · Answer

Answer 1 of 2

Using polynomial long division.

#x-5# Given: #(x^2-2x-15) -:(x+3)# #color(white)("ddddddd.dd") x^2-2x-15# #color(magenta)(x)(x+3)-> ul(x^2+3x larr" Subtract")# #color(white)("dddddddddd") 0 color(white)(",") -5x-15# #color(magenta)(-5)(x+3)-> color(white)("d") ul(-5x-15 larr" Subtract")# #color(white)("dddddddddddddd") 0+0# #(x^2-2x-15) -:(x+3) = color(magenta)(x-5)#

Hazel Anglin · Answer

undefined To divide (x^2 - 2x - 15) by (x + 3) using polynomial long division, follow these steps:

1. Write the dividend (x^2 - 2x - 15) and the divisor (x + 3) in long division format.
2. Divide the first term of the dividend (x^2) by the first term of the divisor (x). The result is x.
3. Multiply the divisor (x + 3) by the result obtained in step 2 (x), and write the product (x^2 + 3x) below the dividend.
4. Subtract the product obtained in step 3 from the dividend. (x^2 - 2x - 15) - (x^2 + 3x) = -5x - 15.
5. Bring down the next term from the dividend (-5x) and rewrite it next to the result obtained in step 2 (x).
6. Divide the new expression (-5x) by the first term of the divisor (x). The result is -5.
7. Multiply the divisor (x + 3) by the result obtained in step 6 (-5), and write the product (-5x - 15) below the previous subtraction.
8. Subtract the product obtained in step 7 from the expression (-5x - 15). (-5x - 15) - (-5x - 15) = 0.
9. There is no remainder, and the division is complete. The quotient is x - 5.

Therefore, (x^2 - 2x - 15)/(x + 3) = x - 5.