- Polynomials are mathematical expressions with one or more terms separated by an addition or subtraction operator.
- In a polynomial, the variables are raised to a non-negative integer power.
- For example, is a polynomial in the variable with three terms.
- The highest power of the variable in any polynomial is called its degree.
- In the above example, the degree of the polynomial is two.
- Note: A polynomial with a single term is called a Monomial.
- The addition & subtraction of polynomials can be carried out in the following two steps:
- Identify like terms in the polynomial
- Combine them according to the correct integer operation (addition or subtraction)
- Note that like terms have the same variables raised to the same exponents.
- The addition or subtraction of polynomials results in a polynomial of the same degree.
- To multiply a polynomial by a monomial, apply the distributive property and simplify each term.
- For example:
- Likewise, to multiply two polynomials, we distribute each term of the one polynomial to all the terms of the other polynomial (FOIL method).
- For example:
- Note: Two or more polynomials, when multiplied, always result in a polynomial of a higher degree (unless one of them is a constant polynomial).
- To divide a monomial by a monomial, divide the numerical coefficients, divide the variables separately, and then multiply the results.
- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
- The long division method is used to divide a polynomial by a polynomial.
Question 1: Add the following polynomials: .
Question 2: Find the product: .
Question 3: Find the product: .
Question 4: Divide by .
Question 5: Find the degree of the polynomial .
Solution: The highest power of the variable in the given polynomial is three; hence, its degree is three.
- To add or subtract the polynomials, add or subtract the like terms.
- To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial.
- The "FOIL" method is used to multiply two binomials.
- To divide a polynomial with a monomial, divide each polynomial term by the monomial.
- To divide two polynomials, use the long division method.
- Below are some special product rules:
- Only like (similar) terms of polynomials can be added or subtracted.
- One must be careful while multiplying polynomials involving negative signs.