How do you divide #(12x^4+12x^3-5x^2+9x+3)/(x^2-4) #?

Question

How do you divide #(12x^4+12x^3-5x^2+9x+3)/(x^2-4)  #?

Nathan Axel · Accepted Answer

The answer is #=12x^2+12x+43+(57x+175)/(x^2-4)#

Let's divide something long.

#color(white)(aaaa)##12x^4##color(white)(aaaa)##12x^3##color(white)(aaaa)##-5x^2##color(white)(aaaa)##9x##color(white)(aaaa)##3##∣##x^2-4#

#color(white)(aaaa)##12x^4##color(white)(aaaaaaaa)####color(white)(aaaa)##-48x^2##color(white)(aaaa)####color(white)(aaaaaa)####∣##12x^2+12x+43#

#color(white)(aaaaaa)##0##color(white)(aaaaa)##12x^3##color(white)(aaa)####color(white)(aaaaaaaaaa)##9x##color(white)(aaaaaa)###

#color(white)(aaaaaa)####color(white)(aaaaaa)##12x^3##color(white)(aaaa)##+43x^2##color(white)(aa)##-48x##color(white)(aa)##+3#

#color(white)(aaaaaa)####color(white)(aaaaaa)##0##color(white)(aaaaaaa)##+43x^2##color(white)(aa)##+57x-172##color(white)(aaaaaa)###

#color(white)(aaaaaa)####color(white)(aaaaaa)####color(white)(aaaaaaaaaaaa)##0##color(white)(aa)##+57x+175##color(white)(aaaaaa)###

Consequently,

#(12x^4+12x^3-5x^2+9x+3)/(x^2-4)#

#=12x^2+12x+43+(57x+175)/(x^2-4)#

Isaiah Anthony · Answer

undefined To divide (12x^4+12x^3-5x^2+9x+3) by (x^2-4), you can use long division. Here are the steps:

1. Divide the first term of the numerator (12x^4) by the first term of the denominator (x^2). The result is 12x^2.
2. Multiply the entire denominator (x^2-4) by the result from step 1 (12x^2), and subtract it from the numerator (12x^4+12x^3-5x^2+9x+3).
   This gives you: (12x^4+12x^3-5x^2+9x+3) - (12x^2 * (x^2-4)).
   Simplifying this expression gives: 12x^4+12x^3-5x^2+9x+3 - (12x^4-48x^2).
   Combining like terms gives: 12x^4+12x^3-5x^2+9x+3 - 12x^4+48x^2.
   Simplifying further gives: 12x^3+43x^2+9x+3.
3. Repeat steps 1 and 2 with the simplified expression (12x^3+43x^2+9x+3) as the new numerator.
   Divide the first term of the new numerator (12x^3) by the first term of the denominator (x^2). The result is 12x.
   Multiply the entire denominator (x^2-4) by the result from step 3 (12x), and subtract it from the new numerator (12x^3+43x^2+9x+3).
   Simplify the expression obtained from the subtraction.
4. Repeat steps 1 and 2 with the new simplified expression as the new numerator.
   Continue this process until you have divided all terms and no further division is possible.
5. The final result will be the quotient obtained from the division.

Note: It is important to keep track of the powers of x and arrange the terms in descending order during the process.