How do you divide #(x^3 - 12x^2 -5x + 6)/(x-2)#?

Question

Kennedy Adams · Accepted Answer

#x^2-10x-25-44/(x-2)# #"one way is to use the divisor as a factor in the numerator"# #"consider the numerator"# #color(red)(x^2)(x-2)color(magenta)(+2x^2)-12x^2-5x+6# #=color(red)(x^2)(x-2)color(red)(-10x)(x-2)color(magenta)(-20x)-5x+6# #=color(red)(x^2)(x-2)color(red)(-10x)(x-2)color(red)(-25)(x-2)color(magenta)(-50)+6# #=color(red)(x^2)(x-2)color(red)(-10x)(x-2)color(red)(-25)(x-2)-44# #rArr"quotient "=color(red)(x^2-10x-25)," remainder "=-44# #rArr(x^3-12x^2-5x+6)/(x-2)=x^2-10x-25-44/(x-2)#

Eli Assouline · Answer

undefined To divide (x^3 - 12x^2 - 5x + 6) by (x - 2), we can use polynomial long division.

First, divide the highest degree term of the numerator (x^3) by the highest degree term of the denominator (x). This gives us x^2.

Next, multiply the entire denominator (x - 2) by the quotient we just found (x^2), and subtract the result from the numerator (x^3 - 12x^2 - 5x + 6). This gives us a new polynomial: -10x^2 - 5x + 6.

Now, repeat the process with the new polynomial (-10x^2 - 5x + 6) and the denominator (x - 2). Divide the highest degree term (-10x^2) by the highest degree term (x), which gives us -10x.

Multiply the entire denominator (x - 2) by the new quotient (-10x), and subtract the result from the new polynomial (-10x^2 - 5x + 6). This gives us a new polynomial: 15x + 6.

Repeat the process again with the new polynomial (15x + 6) and the denominator (x - 2). Divide the highest degree term (15x) by the highest degree term (x), which gives us 15.

Multiply the entire denominator (x - 2) by the new quotient (15), and subtract the result from the new polynomial (15x + 6). This gives us a remainder of 30.

Therefore, the division of (x^3 - 12x^2 - 5x + 6) by (x - 2) is equal to x^2 - 10x + 15 with a remainder of 30.