Polynomials

Here we will talk about adding, subtracting, and multiplying polynomials (including binomials and monomials) that contain one or more variables

The Essentials

Polynomials fit into this form: A x n + B x n - 1 + ... + C x + D . Special cases of the polynomial are the monomial: A x and the binomial: A x + B . Polynomials can also have more than one variable: A x + B x y + C y + D . Polynomials can be added and subtracted by combining like terms.

5 x + 6 + ( x 2 + 12 x − 4 ) = x 2 + 17 x + 2
( 5 x − x y + 2 y − 5 ) − ( 6 x y + 7 ) = 5 x − 7 x y + 2 y − 12

When multiplying a polynomial by a constant or monomial, we perform multiplication with each term in the polynomial by the constant or monomial:

5 ( x 2 + 3 x + 20 ) = 5 x 2 + 15 x + 100
3 x 2 ( 5 x 3 + 4 ) = 15 x 5 + 12 x 2

When multiplying two polynomials together, we multiply each term in one of the polynomials by the entire other polynomial and add them together

( x + 4 ) ( x − 5 )
= x ( x − 5 ) + 4 ( x − 5 )
= x 2 − 5 x + 4 x − 20
= x 2 − x − 20

Example

Multiply these two polynomials:

( x + 7 x y − y ) ( x + 2 )
= x ( x + 2 ) + 7 x y ( x + 2 ) − y ( x + 2 )
= x 2 + 2 x + 7 x 2 y + 14 x y − x y − 2 y
= x 2 + 2 x + 7 x 2 y + 13 x y − 2 y

Practice

Evaluate these polynomial expressions:

  1. ( x + 7 ) + ( x − 5 )

  2. 5 x ( x − 5 x y + y )

  3. ( x + 7 ) ( y − 5 ) + ( x y − 4 ) ( x + 5 )

Solutions:

  1. 2 x + 2

  2. 5 x 2 − 25 x 2 y + 5 x y

  3. x 2 y + 6 x y − 10 x + 7 y − 55

More Resources

Khan Academy: Polynomial Expressions