How do you divide #(6x^4+9x^3-2x^2+2x-7)/(4x-2) #?

Question

How do you divide #(6x^4+9x^3-2x^2+2x-7)/(4x-2)  #?

Avery Alexander · Accepted Answer

Use either synthetic division or polynomial long division to get
#3/2x^3+3x^2+x+1# with remainder of #(-5)# synthetic division #{: (,,6,+9,-2,+2,-7), (+,,,color(white)("X")3,color(white)("X")6,color(white)("X")2,color(white)("X")2), (,,"----","----","---","----","---"), (/ (4),"|",6,12,4,4,color(red)((-5))), (xx (2),"|",color(blue)(3/2),color(blue)(3),color(blue)(1),color(blue)(1),) :}# Polynomial long division #{: (,,color(blue)(3/2x^3),color(blue)(+3x^2),color(blue)(+x),color(blue)(+1),), (,,"----","----","----","----","----"), (4x-2,")",6x^4,+9x^3,-2x^2,+2x,-7), (,,6x^4,-3x^3,,,), (,,"----","-----",,,), (,,,12x^3,-2x^2,,), (,,,12x^3,-6x^2,,), (,,,"-----","-----",,), (,,,,4x^2,+2x,), (,,,,4x^2,-2x,), (,,,,"-----","-----",), (,,,,,4x,-7), (,,,,,4x,-2), (,,,,,"----","----"), (,,,,,,color(red)(-5)) :}#

Eva Abramson · Answer

undefined To divide (6x^4+9x^3-2x^2+2x-7) by (4x-2), you can use long division. Here are the steps:

1. Divide the first term of the numerator (6x^4) by the first term of the denominator (4x). The result is 1.5x^3.

2. Multiply the entire denominator (4x-2) by the result from step 1 (1.5x^3). This gives you 6x^4 - 3x^3.

3. Subtract the result from step 2 (6x^4 - 3x^3) from the original numerator (6x^4+9x^3-2x^2+2x-7). This gives you 12x^3 - 2x^2 + 2x - 7.

4. Bring down the next term from the original numerator (-2x^2). The new numerator becomes 12x^3 - 2x^2 + 2x - 7.

5. Repeat steps 1-4 with the new numerator (12x^3 - 2x^2 + 2x - 7) and the original denominator (4x-2).

6. Continue this process until you have divided all terms of the numerator.

The final result of the division is 1.5x^3 - 0.5x^2 + 1.5x - 3.5, with a remainder of 0.