Operations with MonomialsOperations with Polynomials

Monomial• A number, a variable or a product of a number

and variable(s)

Examples: 1 -15 ½ x A y3

2x 5xyz -3x2 4xy3 ¼ x3

Not Monomials 2x + 4 4x – 5 8

x

6

4

x Isn’t this the same as

61( )

4x

Quick ReviewQuick Review

34xbase

Exponent or power

3 = (4)(x)(x)(x)

the coefficent is not taken to t

What does it mean to raise to a power?

Example: 4x

he power

the x is multiplied times itself 3 times

Coefficient

The Laws of Exponents:The Laws of Exponents:

Multiplicative Law of Exponents: When multiplying monomials and the bases are the same: multiply the coefficients and add the exponents .

m n m nx x x 3 4 3 4 7Example: x x x x

x x x x x x x 7x

Here is why this works…lets break the example down

Example #1(3x2)(-6x3)

-18

Example #2(4y2z2)(y3)

1. Multiply the coefficients2. Write down the bases in alphabetical order

x

3. Add the exponents of the like bases

5

1. Multiply the coefficients

4

2. Write down the bases in alphabetical order

zy

3. Add the exponents of the like bases

5

4. Bring down any other exponents

4. Bring down any other exponents

2

Practice

The Laws of Exponents:The Laws of Exponents:

Exponential Law of Exponents: to raise a monomial to a power - raise the coefficient to the power - Multiply the exponents

323 x

627x

2 2 2(3 )(3 )(3 )x x x

3 2 2 2(3 )( )( )( )x x x

The Laws of Exponents:The Laws of Exponents:

Exponential Law of Exponents: to raise a monomial to a power - raise the coefficient to the power - Multiply the exponents

42 32x y

16

1. 2 to the 4th power = 16

2. Write the variables down

3. Multiply each variables exponent time the exponent on the outside of the parentheses

yx8 12 yx

Practice

The Laws of Exponents:The Laws of Exponents:Division Law of Exponents: When dividing monomials: - simplify the coefficients - subtract the exponents with same bases - Simpify any Negative or Zero exponents

3

2

6 6

3 3

x x x x

x x x

2x

The Laws of Exponents:The Laws of Exponents:Division Law of Exponents: When dividing monomials: - simplify the coefficients - subtract the exponents with same bases - Simpify any Negative or Zero exponents

2

4

2 2

3 3

y x x

y x x x x

2

2

3x

2

5

8x y

4x y

2

3x 0y3

2

x3 2

3 7

12x y

3x y

4

0x 5y

5

4

y5

8

4xz

16yz

1

4

x 3z

3

x

4z

Practice

Binomial – 2 monomials connected by addition or subtraction

Trinomial – 3 monomials connected by addition or subtraction

Polynomial – 1 or more monomials connected by addition or subtraction

2 2x 24 5x y

22 2 5x x

2 2x 22 2 5x x 4 37 3 5 9x x x

Like Terms – have the same variable and the same exponent

Add Polynomials – ignore the parentheses and combine like terms

1.D2 2 2(3 2 5 ) (4 4 3 )x x x x x y 2 2 23 2 5 4 4 3x x x x x y

1. Combine the term with the biggest exponent2. Then combine the terms with next biggest exponent3. Continue combing terms in descending order

2x x 3y

Subtract Polynomials – distribute the negative 1 and combine like terms

7.D 2 2 2(3 2 5 ) (4 4 3 )x x x x x y 24 4 3x x y

1. Distribute the negative 1 (turn everything to its opposite)2. Cross out the 2nd parentheses of original problem3. Combine like terms, going in descending order

29x 9x 3y

1

Add Polynomials – ignore the parentheses and combine like terms

1. Combine the term with the biggest exponent2. Then combine the terms with next biggest exponent3. Continue combining terms in descending order

Subtract Polynomials – distribute the negative 1 and combine like terms

1. Distribute the negative 1 (turn everything to its opposite)2. Cross out the 2nd parentheses of original problem3. Combine like terms, going in descending order