Monomials and adding or subtracting polynomials

A monomial is a number, a variable or a product of a number and a variable where all exponents are whole numbers. That means that

$$42, \: 5x, \: 14x^{12}, \: 2pq$$

all are examples of monomials whereas

$$4+y, \: \frac{5}{y}, \: 14^{x}, \: 2pq^{-2}$$

are not since these numbers don't fulfill all criteria.

The degree of the monomial is the sum of the exponents of all included variables. Constants have the monomial degree of 0.

If we look at our examples above we can see that

Monomial Degree
42 0
5x 0 + 1 = 1
14x12 0 + 12 = 12
2pq 0 + 1 + 1 = 2

A polynomial as oppose to the monomial is a sum of monomials where each monomial is called a term. The degree of the polynomial is the greatest degree of its terms. A polynomial is usually written with the term with the highest exponent of the variable first and then decreasing from left to right. The first term of a polynomial is called the leading coefficient.

$$4x^{5}+2x^{2}-14x+12$$

Polynomial just means that we've got a sum of many monomials. If we have a polynomial consisting of only two terms we could instead call it a binomial and a polynomial consisting of three terms can also be called a trinomial.


Example

What's the degree of the following polynomials?

$$x^{2}+x$$

The first monomial has a degree of 2 and the second monomial has a degree of 1. The highest degree is 2 which mean that the degree of the polynomial is 2.

$$x^{4}+x^{2}+x$$

4, 2 and 1 , the highest degree is 4 which mean that the degree of the polynomial is 4.


We can add and subtract polynomials. We just add or subtract the like terms to combine the two polynomials into one.


Example

Add the polynomials.

$$({\color{green} {4x+8}})+({\color{blue} {3x+2}})$$

We take away the parentheses and group all like terms.

$${\color{green} {4x+8}}+{\color{blue} {3x+2}}$$

$${\color{green} {4x}}+{\color{blue}{3x}}+{\color{green}{ 2}}+{\color{blue} {8}}$$

We add all like terms to get the sum of the polynomials.

$$7x+10$$


Example

Subtract the polynomials.

$$({\color{green}{ 4x+8}})-({\color{blue} {3x+2}})$$

We remove the parentheses and since we got a negative sign before the second parenthesis we need to change the signs.

$${\color{green} {4x+8}}-{\color{blue}{ 3x-2}}$$

$${\color{green} {4x}}-{\color{blue} {3x}}-{\color{green} {2}}+{\color{blue} {8}}$$

Now we subtract all like terms to find the difference between the two polynomials.

$$x+6$$


Video lesson

Add the two polynomials

$$(x^{2}+3x+8)+(3x^{2}-2x+4)$$