5.4. Polynomials

A polynomial is the addition or subtraction of monomials. 

• If we only have two monomials, we called it binomial. (ie. x+12y)
• and, If we have three, we call it trinomial. (ie. x+12y+zt)

The  degree of a polynomial is the biggest degree of its monomials (terms).

More Examples:

• Give the degree of the following polynomial:  2x⁵ – 5x³ – 10x + 9

This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.

This is a fifth-degree polynomial.

• Give the degree of the following polynomial:  7x⁴ + 6x² + x

This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term.

This is a fourth-degree polynomial.

Names of Degrees: When we know the degree we can also give it a name!

Exercise: 

1. The degree of the polynomial x⁵ + 7xy + 3 is ______.
2. The degree of the polynomial y + 7xyz  is ______.
3. The degree of the polynomial x⁵y⁴ + 5x+y + 16 is ______.
4. The degree of the polynomial y⁴ + 7y + 5 is ______.
5. The degree of the polynomial x²y³z⁴ + 7x²y³z³ - 3x¹²y³ is ______.

Solutions: a) Trinomial of degree 5; b) Binomial of degree 3; c)Polynomial of degree 9;d)Trinomial of degree 4;e) Polynomial of degree 15

Evaluating a polynomial is the same as evaluating anything else; you plug in the given value of x, and figure out what y is supposed to be. 

• Evaluate 2x³ – x² – 4x + 2  at  x = –3

I need to plug in "–3" for the "x", remembering to be careful with my parentheses and the negatives:

2(–3)³ – (–3)² – 4(–3) + 2   

=  2(–27) – (9) + 12 + 2   

=  –54 – 9 + 14   

=  –63 + 14   =  –49

Always remember to be careful with the minus signs!