Divide...
1. x^3+2x^2+x+12/x+3
2. 45a^6b^10/5a^3b^2
Note that x^3+2x^2+x+12 is zero for
x = -3, therefore
(x^3+2x^2+x+12)/(x+3) has no remainder. You can divide using long division, but this is not the preferred method if you are doing the calculation by hand. If you make an error, you'll usually have to start all over again. The following method is more transparant. We put:
x^3+2x^2+x+12 = (a x^2 + b x + c) (x+3)
Work out the right hand side:
(a x^2 + b x + c) (x+3) =
a x^3 + (3 a + b) x^2 + (c + 3 b)x + 3 c
Equate the coefficients of the powers of x on both sides.
Equating coefficient of x^3 yields:
a = 1 (1)
Equating coefficient of x^2 yields:
3 a + b = 2 (2)
Equating coefficient of x yields:
c + 3 b = 1 (3)
Equating constant term yields:
3 c = 12 (4)
Inserting (1) in (2) gives:
b = -1
(4) implies that c = 4 and we see that (3) is already satisfied. The fact that there is an extra equation that is automatically satisfied has to do with the fact that the remainder is zero. If you write down the equation:
x^3+2x^2+x+12 = (a x^2 + b x + c) (x+q)
for arbitrary q, then this can only be valid if the remainder is zero. If you then proceed as above then you can rearrange the four equations into three equations for a b and c and one that says that q^3+2q^2+q+12 = 0, i.e. that q is a zero of the polynomial and that the remainder is thus zero.
To divide the expression x^3+2x^2+x+12 by x+3, we can use long division.
Step 1: Write the dividend (x^3+2x^2+x+12) and the divisor (x+3) in long division format:
___________________
x+3 | x^3 + 2x^2 + x + 12
Step 2: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x), which equals x^2. Place this result above the line:
x^2
___________________
x+3 | x^3 + 2x^2 + x + 12
Step 3: Multiply the divisor (x+3) by the quotient obtained in the previous step (x^2). Place the result beneath the dividend:
x^2
___________________
x+3 | x^3 + 2x^2 + x + 12
- (x^3 + 3x^2)
Step 4: Subtract the result obtained in step 3 from the dividend:
x^2 - (x^3 + 3x^2) = -x^3 - x^2
Now we have:
-x^3 - x^2 + x + 12
Step 5: Bring down the next term of the dividend (x):
-x^3 - x^2 + x + 12
x
__________________
x+3 | -x^3 - x^2 + x + 12
- (x^3 + 3x^2)
Step 6: Divide the leading term of the new dividend (-x^3) by the leading term of the divisor (x), which equals -x^2. Place this result above the line:
-x^2
__________________
x+3 | -x^3 - x^2 + x + 12
- (x^3 + 3x^2)
Step 7: Multiply the divisor (x+3) by the new quotient (-x^2). Place the result beneath the new dividend:
-x^2
___________________
x+3 | -x^3 - x^2 + x + 12
- (x^3 + 3x^2)
- (-x^3 - 3x^2)
Step 8: Subtract the result obtained in step 7 from the new dividend:
-x^2 - (-x^3 - 3x^2) = 2x^2 + x
Now we have:
2x^2 + x + 12
Step 9: Bring down the next term of the dividend (12):
2x^2 + x + 12
12
__________________
x+3 | 2x^2 + x + 12
- (2x^2 + 6x)
Step 10: Divide the leading term of the new dividend (2x^2) by the leading term of the divisor (x), which equals 2x. Place this result above the line:
2x
____________________
x+3 | 2x^2 + x + 12
- (2x^2 + 6x)
Step 11: Multiply the divisor (x+3) by the new quotient (2x). Place the result beneath the new dividend:
2x
_____________________
x+3 | 2x^2 + x + 12
- (2x^2 + 6x)
- (2x^2 + 6x)
Step 12: Subtract the result obtained in step 11 from the new dividend:
2x - (2x^2 + 6x) = -5x + 12
Now we have:
-5x + 12
Step 13: We have reached the end of the long division process. The quotient is x^2 - x + 2 and the remainder is -5x + 12.
Therefore,
(x^3 + 2x^2 + x + 12) / (x + 3) = x^2 - x + 2 with a remainder of -5x + 12.
For the second expression,
45a^6b^10 / 5a^3b^2 can be simplified by dividing the coefficients (numbers) and dividing the variables with exponents.
45/5 = 9
a^6 / a^3 = a^3 (subtract the exponents)
b^10 / b^2 = b^8 (subtract the exponents)
Therefore,
45a^6b^10 / 5a^3b^2 = 9a^3b^8.