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Operation | Methods | Result |
---|---|---|
Addition | Combining Like Terms | Polynomial |
Subtraction | Combining Like Terms | Polynomial |
Multiplication | The FOIL Method The Box Method The Distributive Property |
Polynomial |
This lesson completes the set of the four basic operations for polynomials by investigating the methods to divide two polynomials.
Here are a few recommended readings before getting started with this lesson.
Divide the polynomials using long division.
The polynomial long division is a powerful method that helps dividing any two given polynomials. However, when the divisor is a binomial of the form x−k, there is a shortcut that can be applied.
If a polynomial P(x) is divided by a binomial of the form x−k, then the remainder of the division is equal to P(k).
r=P(k)
LHS⋅(x−k)=RHS⋅(x−k)
Distribute (x−k)
x=k
Subtract term
Zero Property of Multiplication
Rearrange equation
Let P(x) be a polynomial and k a real number. The binomial x−k is a factor of P(x) if and only if P(k)=0.
Notice that P(k)=0 means that k is a zero of P(x). Therefore, the theorem can also be stated as follows.
The Factor Theorem is a special case of the Remainder Theorem and establishes a connection between the zeros of a polynomial and its factors.
Since the theorem is a biconditional statement, the proof will consists of two parts.
x=k
Subtract term
Zero Property of Multiplication
x=3
Calculate power and product
Multiply
Add and subtract terms
Find the remainder of the given polynomial division.
When it comes to dividing polynomials, there are two main methods that can be applied — namely, polynomial long division and synthetic division. The table below lists some pros and cons of each method.
Pros | Cons | |
---|---|---|
Polynomial Long Division | Works for every polynomial division | Involves variables, powers, and many computations |
Synthetic Division | Involves only numbers | Works only when the divisor has the form x−k |
To clarify or simplify the calculations, the minus signs can be removed by distributing them to the parentheses. Also, the remaining terms in the dividend can be hidden in the third, fifth, and seventh rows until the terms are needed. This cleaner look can be seen below.
Now, take a look at the process for computing the same division using synthetic division.