How to Square a Polynomial

By Karl Wallulis
The square of a polynomial contains more terms than the original polynomial.

Taking the square of a polynomial with n terms (a + b + c + ... n) is essentially a more complicated version of FOIL (First, Outside, Inside, Last) for multiplying binomials. It is a common mistake in algebra to write (a + b)^2 = a^2 + b^2; in fact, it is a^2 + 2ab + b^2. The inner term 2ab comes from using FOIL to expand the product of the two terms. Using a similar method for longer polynomials results in several inner terms.

Rewrite the square of the polynomial as the polynomial multiplied by itself. For example, rewrite (x^2 - 2x + 3)^2 as (x^2 - 2x + 3)(x^2 - 2x + 3).

Multiply the first term of the first polynomial by each term in the second polynomial, then add the products together. In the above example, multiply x^2 by x^2, -2x and 3 to get x^4 - 2x^3 + 3x^2.

Multiply the next term of the first polynomial by each term of the second polynomial. Add the products, then add the sum to the polynomial from step 2. In the example, you find -2x^3 + 4x^2 - 6x as the new terms.

Continue multiplying terms in the first polynomial by each term in the second polynomial until going through the entire polynomial. In the above example, the finished polynomial is x^4 - 2x^3 + 3x^2 - 2x^3 + 4x^2 - 6x + 3x^2 - 6x + 9.

Combine like terms to find the simplified version of the polynomial. Each of the terms, except for the highest-degree term (x^4 in the example) and the constant (9 in the example), appears exactly twice in the expanded polynomial. Simplifying the above polynomial yields the solution (x^2 - 2x + 3)^2 = x^4 - 4x^3 + 10x^2 - 12x + 9.

About the Author

Karl Wallulis has been writing since 2010. He has written for the Guide to Online Schools website, covering academic and professional topics for young adults looking at higher-education opportunities. Wallulis holds a Bachelor of Arts in psychology from Whitman College.