How do you divide #(-x^4+5x^3-5x^2-6x-2)/(x^2-4) #?

Question

How do you divide #(-x^4+5x^3-5x^2-6x-2)/(x^2-4)  #?

Kennedy Adams · Accepted Answer

#-x^2+5x-9+(14x-38)/(x^2-4)#

The next step is to try long division because the degree of the numerator is greater than the degree of the denominator, but no matter how hard we try, this rational expression just doesn't factor easily.

Express #x^2-4# as the numbers #1color(white)(aa)0color(white)(aa)4#

Express #-x^4+5x^3-5x^2-6x-2# as #-1color(white)(aa)5color(white)(aa)-5color(white)(aa)-6color(white)(aa)2#

This presents the lengthy division issue.

#color(white)(aaaaaaaaa)-1color(white)(aa)5color(white)(aa)-9# #1color(white)(aa)0color(white)(a)-4|bar(-1color(white)(aa)5color(white)(aa)-5color(white)(a)-6color(white)(aa)2)# #color(white)(aaa1aaaa)-ul(1color(white)(aa)0color(white)(aaaa)4)color(white)(aaaa)darr color(white)(a.)darr# #color(white)(aaaaaaaaaaaaa)5color(white)(aa)-9color(white)(aa)-6# #color(white)(aaaaaaaaaaaaa)ul(5color(white)(aaaa)0color(white)(aa)-20)color(white)(a1)darr# #color(white)(aaaaaaaaaaaaaaaa)-9color(white)(aaaa)14color(white)(a)-2# #color(white)(aaaaaaaaaaaaaaaa)ul(-9color(white)(aaaaa)0color(white)(a.a)36)# #color(white)(aaaaaaaaaaaaaaaaaaaaaaa)14color(white)(a)-38#

Thus, the response is

#-x^2+5x-9+(14x-38)/(x^2-4)#

Eli Assouline · Answer

undefined To divide (-x^4+5x^3-5x^2-6x-2) by (x^2-4), you can use long division. Here are the steps:

1. Divide the first term of the numerator (-x^4) by the first term of the denominator (x^2). The result is -x^2.

2. Multiply the entire denominator (x^2-4) by -x^2, and subtract the result from the numerator. This gives you a new numerator: (5x^3-5x^2-6x-2) - (-x^2)(x^2-4).

3. Simplify the new numerator: 5x^3-5x^2-6x-2 + x^4-4x^2.

4. Repeat the process by dividing the first term of the new numerator (5x^3) by the first term of the denominator (x^2). The result is 5x.

5. Multiply the entire denominator (x^2-4) by 5x, and subtract the result from the new numerator. This gives you a new numerator: (-5x^2-6x-2) - (5x)(x^2-4).

6. Simplify the new numerator: -5x^2-6x-2 - 5x^3+20x.

7. Repeat the process until you have no more terms to divide. Continue dividing the new numerator by the denominator, and simplify the numerator each time.

8. The final result will be the quotient of the division.

Note: It is important to double-check your work and simplify the numerator after each step to avoid errors.