Algebra 1 - Polynomials and Operations


  • 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, x2+7x+10 is a polynomial in the variable x 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.

Addition and Subtraction of Polynomials

  • 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.

Polynomial Multiplication

  • To multiply a polynomial by a monomial, apply the distributive property and simplify each term.
  • For example: 2x·x-5=2x·x-2x·5=2x2-10x
  • 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: 5x-2·x+3=5xx+3-2x+3=5x2+15x-2x-6=5x2+13x-6
  • Note: Two or more polynomials, when multiplied, always result in a polynomial of a higher degree (unless one of them is a constant polynomial).

Dividing Polynomials

  • To divide a monomial by a monomial, divide the numerical coefficients, divide the variables separately, and then multiply the results.
  • Example: 14x67x2=147·x6x2=2x4
  • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
  • Example: 8m5+6m3-3m2-2m=8m5-2m+6m3-2m-3m2-2m=-4m4-3m2+3m2
  • The long division method is used to divide a polynomial by a polynomial.

Solved Examples

Question 1: Add the following polynomials: 5m3-n2+2m3-6n2.

Solution: 5m3-n2+2m3-6n2=5m3-n2+2m3-6n2=5m3+2m3-6n2-n2=7m3-7n2

Question 2: Find the product: m-5·3m+6.

Solution: m-5·3m+6=m·3m+6-5·3m+6=3m2+6m-15m-30=3m2-9m-30

Question 3: Find the product: 5-x2.

Solution: 5-x2=5-x·5-x=5·5-x-x5-x=25-5x-5x+x2=25-10x+x2

Question 4: Divide -18x7+4x5-2x2 by x2.

Solution: -18x7+4x5-2x2x2=-18x7x2+4x5x2--2x2x2=-18x5+4x3+2

Question 5: Find the degree of the polynomial x3-2x2+4.

Solution: The highest power of the variable in the given polynomial is three; hence, its degree is three.

Cheat Sheet

  • 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:
    • x+y2=x2+2xy+y2
    • x-y2=x2-2xy+y2
    • x+yx-y=x2-y2
    • x+y3=x3+3x2y+3xy2+y3
    • x-y3=x3-3x2y+3xy2-y3
    • (x+y)(x2-xy+y2)=x3+y3
    • (x-y)(x2+xy+y2)=x3-y3

Blunder Areas

  • Only like (similar) terms of polynomials can be added or subtracted.
  • One must be careful while multiplying polynomials involving negative signs.