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

Question

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

Isaiah Anthony · Accepted Answer

#color(magenta)(-x^2-4x+1# plus remainder of #color(magenta)(20x-6# #(-x^4-4x^3-3x^2+4x-2)/(x^2+4)# #color(white)(..........)color(white)(.)color(magenta)(-x^2-4x+1# #w-2|overline(-4x^4-4x^3-3x^2+4x-2)# #color(white)(..............)ul(-x^4+0x-4x^2)# #color(white)(.....................)-4x^3+x^2+4x# #color(white)(......................)ul(-4x^3+0x-16x-2)# #color(white)(..................................)x^2+20x-2# #color(white)(..................................)ul(x^2+0x+4)# #color(white)(..........................................)color(magenta)(20x-6# #(-x^4-4x^3-3x^2+4x-2)/(x^2+4) = color(magenta)(-x^2-4x+1# plus remainder of #color(magenta)(20x-6#

Genesis Allen · Answer

undefined To divide (-x^4-4x^3-3x^2+4x-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: (-x^4-4x^3-3x^2+4x-2) - (-x^2)(x^2+4).
3. Simplify the new numerator: -4x^3-3x^2+4x-2 + (x^4+4x^2).
4. Repeat the process by dividing the first term of the new numerator (-4x^3) by the first term of the denominator (x^2). The result is -4x.
5. Multiply the entire denominator (x^2+4) by -4x, and subtract the result from the new numerator. This gives you a new numerator: -4x^3-3x^2+4x-2 - (-4x)(x^2+4).
6. Simplify the new numerator: -3x^2+4x-2 + (4x^3+16x).
7. Continue these steps until you have simplified the entire numerator.

The final result of the division is: -x^2 - 4x + 4 + (4x^3+16x)/(x^2+4).