How to Simplify Polynomials

Simplifying polynomials is the first step to solving algebraic equations

Algebra revolves around polynomials, especially the quadratic equation. To be a polynomial, each term must have only whole number exponents on its variable, although a polynomial may have many different variables. In practical application, such as engineering and biology, polynomials can be long and difficult to deal with unless you know how to simplify them. Simplifying polynomials involves sorting the terms and combining them until each term has a different rank or the variables in terms of the same rank are different.

Instructions

1. Sort the terms by rank, from highest to lowest. The exponent tells you the rack of each term if it has only one variable. You must add the exponents of all the variables in a multi-variable term. Constants are rank zero.

   Example
   2(x^3)(y) + 2(x^4)(y) + 3(x)(y^3) - 6(x^2)(y^3) + 5(x^2)(y^2) - 8(y^4) + 3(x^3)(y^2) + x^5 - 17(x^2)(y^2) +6(x)(y^3) -10(y^4) - 6(x^4)(y) + (x^3)(y) -10 = 0

   The first term is rank "4", because the x-exponent (3) and the y-exponent (1) add up to four. The second term is rank "5" because the exponents add to five. In order from highest rank to lowest rank, the polynomial becomes:

   2(x^4)(y) - 6(x^2)(y^3) + 3(x^3)(y^2) - 6(x^4)(y) + x^5 + 2(x^3)(y) + 3(x)(y^3) + 5(x^2)(y^2) - 8(y^4) - 17(x^2)(y^2) + 6(x)(y^3) -10(y^4) + (x^3)(y) -10 = 0

2. Sort again ensuring that at least one variable is in rank order from highest to lowest. For single variable polynomials, this step is unnecessary. In this example, the term "3(x^3)(y^2)" would come before the term "- 6(x^2)(y^3)." Both are rank "5", but the x-variable rank in the first term is "3," while the x-variable rank in the second term is "2." The example would look like this after the second sorting.

   x^5 + 2(x^4)(y) - 6(x^4)(y) + 3(x^3)(y^2) - 6(x^2)(y^3) - 8(y^4) -10(y^4) + 2(x^3)(y) + (x^3)(y) + 5(x^2)(y^2) - 17(x^2)(y^2) + 3(x)(y^3) + 6(x)(y^3) -10 = 0

3. Combine all the terms of the same rank by adding their exponents. For multi-variable polynomials, each variable in the expression must have the same rank individually as well. In this example, you can combine "5(x^2)(y^2)" with "- 17(x^2)(y^2)" because both terms are rank "4" and the x-exponent and the y-exponent in both is "2." The polynomial after all allowable combinations is:

   x^5 - 4(x^4)(y) + 3(x^3)(y^2) - 6(x^2)(y^3) + 3(x^3)(y) - 12(x^2)(y^2) + 9(x)(y^3) - 18(y^4) -10 = 0

   The term "-18(y^4)" is the same as "- 18(y^4)(x^0)" and the term "-10" is the same as "-10(x^0)." Therefore, both terms x-exponent rank is "0".

Tips & Warnings

• You can further simplify some polynomials by factoring them. Factoring the example polynomial yields: [x^2 + 3y][x^3 - 4(x^2)(y) + 3(x)(y^2) - 6(y^3)] =10

References

• Paul's Online Math Notes: Algebra: Polynomials; Lamar University

Resources

• Paul's Online Math Notes: Algebra; Lamar University