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Polynomials and Radicals Help

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Updated on Oct 27, 2011

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 y 5xy2 19a6b4

Polynomials

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

7y5 –6ab4 8x + y3 8x + 9yz

Kinds of Polynomials

Here are three kinds of polynomials:

  1. A monomial is a polynomial with one term, such as 5b6.
  2. A binomial is a polynomial with two unlike terms, such as 2x + 4y.
  3. A trinomial is a polynomial with three unlike terms, such as y3 + 8z – 2.

Operations With Polynomials

To add polynomials, simply combine like terms. Let's look at (5y3 – 2y + 1) + (y3 + 7y – 4).

First remove the parentheses: 5y3 – 2y + 1 + y3 + 7y – 4. Then arrange the terms so that like terms are grouped together: 5y3 + y3 – 2y + 7y + 1 – 4. Now combine like terms: 6y3 + 5y – 3. Just to make sure you've got it, let's look at another example.

Examples

Example: 1

(2x – 5y + 8z) – (16x + 4y – 10z)

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

2x – 5y + 8z – 16x – 4y + 10z

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

2x – 16x – 5y – 4y + 8z + 10z

Now combine like terms:

–14x – 9y + 18z

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

Example 2:

(–4a3b)(6a2b3) = (–4)(6)(a3)(a2)(b)(b3) = –24a5b4

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:

8x(12x – 3y + 9)

Distribute. (8x)(12x) – (8x)(3y) + (8x)(9)
Simplify. 96x2 – 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 – 3y + 7

FOIL

The FOIL method is used when multiplying binomials. FOIL represents the order used to multiply the terms: First, Outer, Inner, and Last. To multiply binomials, you multiply according to the FOIL order and then add the products.

Example

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

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

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

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

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

Now multiply according to FOIL:

(4x)(9x) + (4x)(8) + (2)(9x) + (2)(8) = 36x2 + 32x + 18x + 16

Now combine like terms:

36x2 + 50x + 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) = 3x + 3y

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

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