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

Question

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

Abigail Allen · Accepted Answer

#x^3+5x-14+26/(x+2)#

I'd like to use synthetic division for #color(red)(1)x^4+color(blue)(2)x^3+color(purple)(5)x^2+color(green)(-4)x+color(orange)(-2)#

#color(black)(-2)color(black)(|)color(red)(1)color(white)(-)color(blue)(2)color(white)(-)color(purple)(5)color(white)(.)color(green)(-4)color(white)(.)color(orange)(-2)# #color(white)(-2)color(black)(|)color(white)(1)color(white)(.)color(black)(-2)color(white)(-)color(black)(0)color(white)(.)color(black)(-10)color(white)(.)color(black)(28)# #color(white)(-2.)..........................................# #color(white)(-2)color(black)(|)color(black)(1)color(white)(-)color(black)(0)color(white)(-)color(black)(5)color(white)(.)color(black)(-14)color(white)(.)color(black)(26)#

There is a remainder, but it poses no issue.

#x^3+0x^2+5x-14+26/(x+2)# #x^3+5x-14+26/(x+2)#

Eli Assouline · Answer

undefined To divide (x^4+2x^3+5x^2-4x-2) by (x+2), 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). This gives x^3.
2. Multiply the entire denominator (x+2) by the result from step 1 (x^3), and subtract it from the numerator (x^4+2x^3+5x^2-4x-2).
   This gives (x^4+2x^3+5x^2-4x-2) - (x^3(x+2)).
3. Simplify the expression obtained in step 2.
4. Repeat steps 1-3 with the simplified expression until you have no more terms to divide.

By following these steps, you can continue dividing until you reach the end of the polynomial.