How do you divide #(x^4+2x^3+ 15 x^2+6x+8)/(x-4) #?

Question

How do you divide #(x^4+2x^3+ 15 x^2+6x+8)/(x-4)  #?

Savannah Anderson · Accepted Answer

#(x^4+2x^3+15x^2+6x+8)/(x-4)=color(red)(x^3+6x^2+39x+152)" Remainder: "color(blue)(616)#

Use "long division" or "synthetic division" for this division.

synthetic division #{: (,,x^4,x^3,x^2,x^1,x^0," usually omitted"), (,,1,2,15,6,8," the coefficients"), (ul(+),ul(color(white)("|")),ul(0),ul(4),ul(24),ul(146),ul(608)," product of prior row 3 value and "x"'s zero"), (4,"|",1,6,39,152,616," the 4 is the value of "x" that would make the denominator "=0), (,,x^3,x^2,x^1,x^0,R," again, this is usually omitted") :}#

Avery Alexander · Answer

undefined To divide (x^4+2x^3+15x^2+6x+8) by (x-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). This gives x^3.
2. Multiply the entire denominator (x-4) by the quotient obtained in step 1 (x^3). This gives x^3(x-4) = x^4-4x^3.
3. Subtract the result obtained in step 2 from the numerator (x^4+2x^3+15x^2+6x+8) to get the new numerator: (2x^3+15x^2+6x+8) - (x^4-4x^3) = -x^4+6x^3+15x^2+6x+8.
4. Repeat steps 1-3 with the new numerator (-x^4+6x^3+15x^2+6x+8) until the degree of the new numerator is less than the degree of the denominator.
5. The final quotient is the sum of all the quotients obtained in each step. In this case, the quotient is x^3.

Therefore, (x^4+2x^3+15x^2+6x+8)/(x-4) = x^3.