Basic Arithmetic Operations on Polynomials
Adding PolynomialsThe like terms in two polynomials being added together can combine together. In the example below the purple, red, and blue highlighted terms are all respectively like terms. The two purple colored terms can be added together since they both have the same variable. The same method is applied to the red colored terms. The blue term does not have another term that it can combine with from the other polynomial, therefore the term reams at the same value in the final answer.
Subtracting PolynomialsSubtracting polynomials is very similar to adding polynomials in that only the like terms are able to be subtracted from one another. Our purple term share an a^3, the red share the 'b' variable, and the blue term do not have variables.
Multiplying PolynomialsTo multiple polynomials each term from each polynomials must be multiplied by the terms in the other polynomial. There is a method referred to as 'FOIL' that creates a simple four step process for multiplying polynomials that have two terms each.
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Diving PolynomialsLong Division
Long division is the longer method of dividing polynomials. It also is the method that is more likely to be correct across all polynomials. The process of long division is exactly like that of regular long division. The denominator is multiplied to be a number to be equivalent to the first value in the numerator. The amount that is multiplied to get that value is the first term in the quotient. The product is then subtracted from the numerator. the resulting value is used as the process is repeated using it as the next numerator.
Synthetic Division
Synthetic division is the quicker alternative to long division. It will work with most polynomials. The method uses only the values and not the variables. In the example below you can see how the values from the equation were transferred into a table across the top row. In all cases the first value from the numerator is dropped under the line. That term is then multiplied by the opposite of the value from the denominator. The product is placed under the second value from the numerator. These numbers subtract to create the next value that is place below the line. The process is then repeated as the new value is multiplied by the opposite of the value from the denominator and the steps repeat till all values from the numerator have given a value to the quotient. the final value in the quotient section is the remainder that is to be placed over the denominator when writing out the answer using the variables.
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