Multiplying and Dividing Polynomials Multiplying and Dividing Polynomials Polynomials Taught By Eles Jones Just like in any expression , with polynomials, you add or subtract like terms together. Like terms are terms that are alike. For example :

(2x + 13) + ( 3x + 5 )

in this equation , you would add the "like terms " together.

3x+2x = 5x . 13+5 = 18

The new equation would become ,

(2x +13 ) + (3x + 5 ) = (5x+18) Here are three types of polynomials. A monomial , a binomial, and a trinomial . Adding and Subtracting Polynomials , cont. When it comes to subtracting polynomials , there is a bit of a twist. Lets say you had the problem :

(3x + 24) - (2x +15)

When subtracting polynomials , you reverse the sign of the terms behind the minus sign. So the expression would change to :

(3x+24) - (- 2x - 15)

Then you would solve the expression as usual.

3x -2x = x 24 - 15 = 9

Now the answer to the problem would be :

(3x+24) - (2x+15) = (x + 9) Standard Form What is standard form? Standard form is the way that polynomials are written , from the largest to the smallest. In any expression , polynomials are supposed to be lined up by the highest degree. Degrees in polynomials are the exponents. If your polynomial had the numbers 3x , 12 , 15x and 23x , the order of the numbers would be :

3x + 15x + 23x +12

3x is first because it has the highest degree. Even though the constant of the other terms were larger than the constant of 3x , they still did not have the higest degree because constants do not affect the degree. Do It Yourself ! Put the following numbers in standard form.

1. 3x , 12 , 33x , 4 , 123x , 54 ,

2. 22 , 46x , 18x , 23, 987x

3. x , 2x , 3x , 4x , 5x What are polynomials? Adding and Subtracting Polynomials Different Types of Polynomials Adding and Subtracting Polynomials A polynomial is an expression that can contain variables, constants, and exponents but :

- No division by a variable

-a variables exponents can only be 0,1,2,3....

- it can't have an infinite number of

terms. Multiplying and dividing polynomials Examples of Polynomials

- 2x + 5x +3x + 7

- 2x - 15 3 2 NOT Polynomials

- 5xy

-3/6 + 27x A monomial

contains one term.

- 3x

- 45y

-12

-2356x

-72x A binomial

contains two terms.

- 3x + 27

- 45x + 23x

- 12x -15

- 23x + 235

- 3x + 8x 2 A trinomial contains of three terms.

-4x + 3x + 12

-87x + 2x - 27

-86x - 23x - 4

-34x - 2x +123

-x + x - 3 Parts of a Polynomial 2x + 5x +3x + 7 Terms 3 2 Exponents Terms Every part of a polynomial that is being added is a called a term. Terms can contain variables, exponents or constants. There can be no negative exponents, variables in the denominator of the term , or square roots of variables. Examples Non-examples

- 2x - 1/x

- 12 -23x

-64x - xy(z) -2 3 6 5 Try it yourself ! Match each term with the correct meaning. 2x(y) Not a term

3x/2 Term

12 Not a term

23x Term

1/x Not a term

86 Term -2 8 23 2 3 2 When multiplying polynomials, we use whats know as the "Punnett Square Method". In the Punnett square method, we take the two expressions we are multiplying , and set them up into a punnett square. for example , if our problem was (3x + 12) * (2x + 4) . this is how we would set up our square. 3x + 12 2x

+ 4 We would then multiply each term by another from the other expression. For example , 3x times 2x , would equal 6x , because variables multiplied by each other equal that variable squared. Then we would do the same with the other terms. 12 times 2x would equal 24x , 3x times 4 would equal 12x , and 12 times 4 would equal 48. So then our solution would be:

6x + 36x + 48 because you had to add like terms. 2 6 3 2 6 6 3 2 2 3 5 2 5 4 3 6 2 When dividing polynomials, you must use the exponents rule for all the terms. Lets work with this problem :

(12x - 20x + 8) / 4.

First we must divide each term by the monomial. 12x divide by 4 would be 3x since there is no exponent to cancel out. 2 2 Then -20x divided by 4 would equal -5x . Then 8 divided by 4 would become 2. So the full equation would be :

(12x - 20x +8) / 4 = 3x - 5x + 2 2 3 2 2 2 2 Let's Practice ! Solve the following problems. 1. (2x + x - 7) + (x + x + 6)

2. (x - 8) - (4x + 7x)

3.(x - 2) (5 + 3x)

4. (x + 7x - 28) / 7x 2 2 2 2 2 Answer Key 1. 3x + 2x - 1

2. -3x - 7x - 8

3. 3x - x - 10

4. x/7 + 1 - 4/x 2 2 2 2 2 2

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