07/03/2024
Preparing for North Carolina Math-3
Polynomial Division
This is when you divide one algebraic expression by another. It can be helpful/used-by computer programmers to know if there will be a situation when there will be a division by zero error.
There are two methods of performing this task. The traditional method works every time under every circumstance. Synthetic division only works if you are dividing by a linear expression (a variable to the first power with a constant).
Traditional Method
1 - Arrange your term/numerator from the greatest exponent to the constant. If there is a missing exponent, add it with a coefficient of zero.
2 - Do the same thing with the divisor/denominator.
3 - Looking at the first term in the numerator and the first term in the denominator, find the term by which the denominator's term must be multiplied to equal the numerator's term.
4 - Multiply the entire denominator by this term, and subtract it from the numerator.
5 - Use the remaining difference as the new numerator, and repeat steps 3 - 5 until the difference is zero or the term you added to the quotient is a constant. If the latter is true, you have a remainder. Put the remainder atop the divisor as the final part of your quotient.
Example
2x^3 - 3x^2 + 4x -5
----------------------
x^2 + 2
As the first term of the numerator is 2x^3, and the first term of the divisor is x^2, you want to multiply the divisor by 2x.
2x^3 - 3x^2 + 4x - 5
-(2x^3 + 4x)
--------------------------
-3x^2 - 5
Since the first term of the difference is -3x^2, and the first term of the divisor is x^2, you multiply the divisor by -3.
-3x^2 - 5
-(-3x^2 - 6)
---------------
1
Your remainder is 1/(x^2 + 2), and your complete quotient is:
2x - 3 + 1/(x^2 + 2)
Another Example
2x^3 + 3x^2 - 3x - 2
-----------------------
2x + 1
With the first term of the numerator being 2x^3, and the first term of the denominator being 2x, the denominator has to be multiplied by x^2.
2x^3 + 3x^2 - 3x - 2
-(2x^3 + 1x^2)
-------------------------
2x^2 - 3x - 2
With the first term of the numerator being 2x^2, and the first term of the denominator being 2x, the denominator has to be multiplied by x.
2x^2 - 3x - 2
-(2x^2 + 1x)
-----------------
- 4x - 2
With the first term of the numerator being -4x, and the first term of the denominator being 2x, the denominator has to be multiplied by -2.
-4x - 2
-(-4x - 2)
----------
0
There is no remainder, so the quotient is x^2 + x - 2
Synthetic Division
This method only works when dividing by a linear expression. If you have a polynomial in the denominator that can be factored into multiple linear expressions (binomial terms), you can user this method using the quotient and the next binomial.
1 - Arrange your term/numerator from the greatest exponent to the constant. If there is a missing exponent, add it with a coefficient of zero.
2 - Set the linear expression by which you are dividing to zero, and solve for the variable. This becomes your "multiplier."
3 - List the coefficients from your numerator.
4 - Bring-down the left-most coefficient.
5 - Multiply this coefficient by the multiplier, and add it to the next coefficient.
6 - Bring-down the sum, and repeat step 5 until you have added a product to the constant from the original term.
7 - If the final sum is not zero, you have a remainder.
8 - If there was a coefficient in your divisor, divide all terms that you just found by that coefficient.
9 - Decrease the highest exponent from the original numerator by one, and connect these variables to the numbers you just found.
Example
2x^3 + 3x^2 - 3x -2
----------------------
2x + 1
Determine your "multiplier" by setting 2x + 1 = 0. Solving for x, you have x = -1/2.
Your coefficients are:
2 3 -3 -2
Bring-down the 2, and multiply it by -1/2 to get -1.
3 + (-1) = 2
Multiply 2 by -1/2 to get -1.
-3 + (-1) = -4
Multiply -4 by -1/2 to get +2.
-2 + (+2) = 0
Your original numerator had the greatest variable ("degree") of 3; therefore, your first term of your quotient has an x^2.
2x^2 + 2x - 4 with no remainder is NOT your answer, because the denominator had a coefficient of 2; therefore, this answer has to be divided by 2 to get:
x^2 + x - 2
This is the same quotient as the second example using the traditional method.
*** NOTE ***
Step 8 of this method often is skipped by teachers. Don't make this mistake!!
