Steps
2-4 are repeated until the polynomial obtained in step
4 has a
lower degree than the denominator. Dividing the leading terms
-4x2 and
x2 gives
-4 as result. Next, multiply the denominator by
-4 and write the result at the bottom.
x2+2x+53x−4 3x3+2x2+0x−7−(3x3+6x2+15x)-4x2−15x−7-4x2−8x−20
Now, subtract the two polynomials at the bottom.
x2+2x+53x−4 3x3+2x2+0x−7−(3x3+6x2+15x)-4x2−15x−7−(-4x2−8x−20)-7x+13
This time, the resulting polynomial has a degree of
1, which is lower than the denominator's degree. Therefore, the division is complete. This implies that the of the division is
3x−4 and the remainder is
-7x+13.
Q(x)R(x)=3x−4=-7x+13
Finally, the initial polynomial division can be written as the
quotient plus the division of the
remainder and the
denominator.
x2+2x+53x3+2x2−7=3x−4+x2+2x+5-7x+13