A polynomial in one variable is the sum or difference of terms of the form , where is a real number and is a nonnegative integer. Polynomials with one, two, or three terms have their own names.
- A monomial is a polynomial expression consisting of one term. An example of a monomial is .
- A binomial is a polynomial expression consisting of two terms. An example of a monomial is .
- A trinomial is a polynomial expression consisting of three terms. An example of a monomial is .
Adding and Subtracting Polynomials
Two common methods for multiplying polynomials include using different types of properties, including the distributive property and properties of exponents.
To use the distributive property for any two polynomials, distribute the first polynomial to each term in the second polynomial. Repeat this process of distribution as needed, depending on the number of terms in the polynomials. The goal is to multiply each term in the first polynomial by each term in the second polynomial.To use the product of powers property, multiply two powers with the same base by adding the exponents.
Polynomial Factoring Methods
|Greatest common factor (GCF)||When each term has a common factor, divide the greatest common factor from each term.|
|Factor by grouping||This method is most commonly used when there are four terms. Group the terms into two pairs. Then factor out the GCF of each pair of terms. The resulting terms have a common factor that can be factored out.|
|Factor a trinomial||Many trinomials of the form can be written as the product of two binomials. Find two numbers that are factors of and have a sum of . Use these numbers to write the binomial factors or to write a four-term polynomial and continue with factoring by grouping.|
|Perfect square trinomial||Some trinomials fit a pattern:
|Difference of squares||Some binomials fit a pattern with two squared terms:
|Sum or difference of cubes||Some binomials fit a pattern with two cubed terms:
Identify whether the binomial fits a pattern.The binomial is a difference of cubes:
There are different techniques for dividing polynomials, depending on the polynomials that are given. The combination of monomials and polynomials will determine which technique to use. To divide by a monomial, divide each term of the dividend by the monomial. To divide by a polynomial, start by trying to factor the dividend and divisor.The quotient of powers property states: To divide two powers with the same base, subtract the exponents.
Use the values in the bottom row as coefficients for each term in the quotient.The degree of the quotient will be one less than the degree of the original dividend. After the constant term, write the remainder as a fraction with the divisor as the denominator.