**Introduction to Polynomials**

**Monomials**

A monomial is an expression that is a number, a variable, or a product of a number and one or more variables.

6 y5 xy^{2}19 a^{6}b^{4}

**Polynomials**

A polynomial is a monomial or the sum or difference of two or more monomials.

7 y^{5}–6 ab^{4}8 x+y^{3}8 x+ 9y–z

**Kinds of Polynomials**

Here are three kinds of polynomials:

- A
**monomial**is a polynomial with one term, such as 5*b*^{6}. - A
**binomial**is a polynomial with two unlike terms, such as 2*x*+ 4*y*. - A
**trinomial**is a polynomial with three unlike terms, such as*y*^{3}+ 8*z*– 2.

**Operations With Polynomials**

To add polynomials, simply combine like terms. Let's look at (5*y*^{3} – 2*y* + 1) + (*y*^{3} + 7*y* – 4).

First remove the parentheses: 5*y*^{3} – 2*y* + 1 + *y*^{3} + 7*y* – 4. Then arrange the terms so that like terms are grouped together: 5*y*^{3} + *y*^{3} – 2*y* + 7*y* + 1 – 4. Now combine like terms: 6*y*^{3} + 5*y* – 3. Just to make sure you've got it, let's look at another example.

**Examples**

**Example: 1**

(2*x* – 5*y* + 8*z*) – (16*x* + 4*y* – 10z)

First remove the parentheses. Be sure to distribute the subtraction correctly to all terms in the second set of parentheses:

2*x* – 5*y* + 8*z* – 16*x* – 4*y* + 10*z*

Then arrange the terms so that like terms are grouped together:

2*x* – 16*x* – 5*y* – 4*y* + 8*z* + 10*z*

Now combine like terms:

–14*x* – 9*y* + 18*z*

To multiply monomials, multiply their coefficients and multiply like variables by adding their exponents.

**Example 2: **

(–4*a*^{3}*b*)(6*a*^{2}*b*^{3}) = (–4)(6)(*a*^{3})(*a*^{2})(*b*)(*b*^{3}) = –24*a*^{5}*b*^{4}

To divide monomials, divide their coefficients and divide like variables by subtracting their exponents.

**Example 3: **

To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products.

**Example 4: **

8*x*(12*x* – 3*y* + 9)

Distribute. (8 x)(12x) – (8x)(3y) + (8x)(9)Simplify. 96 x^{2}– 24xy+ 72x

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and add the quotients.

**Example 5:**

= *x* – 3*y* + 7

**FOIL**

The FOIL method is used when multiplying binomials. FOIL represents the order used to multiply the terms: **F**irst, **O**uter, **I**nner, and **L**ast. To multiply binomials, you multiply according to the FOIL order and then add the products.

**Example**

(4*x* + 2)(9*x* + 8)

**F:** 4*x* and 9*x* are the **first** pair of terms.

**O:** 4*x* and 8 are the **outer** pair of terms.

**I:** 2 and 9*x* are the **inner** pair of terms.

**L:** 2 and 8 are the **last** pair of terms.

Now multiply according to FOIL:

(4*x*)(9*x*) + (4*x*)(8) + (2)(9*x*) + (2)(8) = 36*x*^{2} + 32*x* + 18*x* + 16

Now combine like terms:

36*x*^{2} + 50*x* + 16

**Figuring Out Factoring**

**Introduction to Factoring**

Factoring is the reverse of multiplication. When multiplying, you find the product of factors. When factoring, you find the factors of a product.

Multiplication: 3(*x* + *y*) = 3*x* + 3*y*

Factoring: 3*x* + 3*y* = 3(*x* + *y*)

### Ask a Question

Have questions about this article or topic? Ask### Related Questions

See More Questions### Popular Articles

- Kindergarten Sight Words List
- First Grade Sight Words List
- 10 Fun Activities for Children with Autism
- Child Development Theories
- Grammar Lesson: Complete and Simple Predicates
- Definitions of Social Studies
- Social Cognitive Theory
- Signs Your Child Might Have Asperger's Syndrome
- Why is Play Important? Social and Emotional Development, Physical Development, Creative Development
- Theories of Learning