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In addition to the revision notes for Operations with Polynomials on this page, you can also access the following Polynomials learning resources for Operations with Polynomials
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 11.2 | Operations with Polynomials |
In these revision notes for Operations with Polynomials, we cover the following key points:
Like terms in a polynomial are those monomials which have the same variable part but their coefficients are different.
Unlike terms in a polynomial are those monomials, which have the different variable parts although their coefficients may be the same.
Polynomials are in the standard form when their terms are written in descending order in regard to variables indices. If written so, the term with the highest degree is called the leading term and the coefficient of this term is called the leading coefficient.
When adding or subtracting polynomials, the only terms that are added or subtracted are the like ones. This is because they represent the same unknown part and we know from arithmetics that we can add or subtract only items that belong to the same category after collecting the like terms.
It is better to write the polynomials in such a form that one variable is taken as a reference and all terms are written in descending order of this variable's power. Then, we do the same thing for the rest of the variables.
In the multiplication of polynomials, it is not necessary to have only like terms in order to multiply them. The general rule is that coefficients are multiplied while the indices (exponents) of the same type of variables are added.
In the multiplication of polynomials, we multiply each term of the first polynomial by each term of the second polynomial. As a special case, when the two polynomials have two terms each, we apply the FOIL rule (FOIL is an acronym that stands for "First - Outside - Inside - Last". This means when we have two linear polynomials P(x) = (ax + b) and Q(x) = (cx + d), we can multiply them in the following way
Hence, the first term of the first polynomial multiplies the first term of the second polynomial (hence the name FIRST in the FOIL Rule); then the first term of the first polynomial multiplies the second (outside) term of the second polynomial (hence the name OUTSIDE in the FOIL Rule). Then, we continue with the multiplication of the second term of the first polynomial and the first term of the second polynomial, i.e. with the terms that are on the INSIDE of the expression and LAST, we multiply the second terms of the two polynomials.
When dividing two monomials, the indices of the same variable in each term are subtracted, obeying the same rules as in the division of numbers written in powers, i.e.
As for the coefficients of the monomials involved, they are divided. In popular terminology, we say, "the monomials are simplified" when doing such an operation.
When a polynomial is divided by a monomial, we have two options for the solution procedure:
Basically, the division of two polynomials has the same logic as the division of numbers, where a number x (dividend) is divided by another number y (divisor) to give a number z (quotient) and a remainder r as a result.
When two polynomials P(x) and Q(x) are divided, the result is another polynomial S(x) and another polynomial R(x) as a remainder. In symbols, we write
The easiest case in the division of polynomials is the division without remainder, where after multiplying the result (quotient) and the divisor gives the original polynomial (dividend).
Raising a polynomial in a given power means multiplying it several times by itself. Hence, the multiplication rules are also applied when a polynomial is raised in a given power.
Polynomials at one variable are suitable to work with, as they give smooth graphs. In more advanced topics such as differentiation and integration, it is very important to have smooth graphs, as it is easier to determine the minimum or maximum of the graph line. If the graph represents a polynomial, these extremums represent the zeroes of the polynomial.
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