Monomial

An algebraic expression with only one term.

This can be a number, a product of numbers or variables with whole number exponents (no negative exponents or fractions allowed)

Examples:

• 4
• 3x³
• xyz
• [image] (these are not monomials!!!)

Binomial

An algebraic expression with two terms.

Examples:

• 3x+2
• x²+21

Trinomial

An algebraic expression with three terms.

Examples:

• 3x²+5x-5
• 5x⁴+3x+1

Polynomial

Is a polynomial that has MORE than three terms.

Examples:

• x⁴-x³+5x-2
• x³-x²+x+32

Constant

This is the number that never changes.

It always stays constant.

It is never seen with a variable next to it!

x²+5x-2

x²+5x-2

-2 is the constant term

Standard form of a polynomial

This is when a polynomial is written with the terms in order from the greatest degree to the least degree.

Example:

3x³-x+5x²+7x⁵  - Not in standard form

7x⁵+3x³+5x²-x  - In standard form

Leading Coefficient

The number that is next to the term with the greatest degree when in standard form.

Example:

-7x³-3x²-2x+4

-7 is the Leading coefficient!

4 is the constant term!

A polynomial that has a degree of zero

is called a?

Constant

3

3⁰ or x⁰

A polynomial that has a degree of one

is called?

Linear

x or 2x-1

x¹ or 2x¹-1

Warning this is not a linear polynomial:

9³+x⁰

A polynomial that has a degree of two

is called?

Quadratic

Examples:

x² or 2x²+x

Warning this is not a quadratic polynomial:

7²+x

A polynomial that has a degree of three

is called?

Cubic

x³

Warning this is not a cubic polynomial:

5³+x

A polynomial that has a degree of four

is called?

Quartic

Cubic

x⁴

Warning this is not a cubic polynomial:

5⁴+x

A polynomial that has a degree of five

is called?

Quintic

x⁵

Warning this is not a cubic polynomial:

7⁵+x

A polynomial that has a degree of six

is called?

6th degree

x⁶

In fact any polynomial greater than 6 will be related to this way!

x⁷ is a 7th degree polynomial etc...

Warning this is not a sixth degree polynomial:

7⁶+x

Find the degree of this:

Monomial: -2a²b⁴

To find the degree of any monomial

we simply add their exponents!

-2a²b⁴

-2x⁰a²b⁴   

(recall that a constant has a degree of 0)

0+2+4=6

This is a 6th degree monomial!

Find the degree of this polynomial:

12x-4x³+2-x²

First put this beast in standard form (ordering the exponents from greatest to the least)

-4x³-x²+12x+2

This is a cubic!

-4 is the leading coefficient

and 2 is the contstant term

Classify this polynomial according to its degree

and number of terms

6x²-x³

First we need to put this in standard form:

-x³+6x²

This is a cubic and has two terms which is called

a binomial. Therefore this is called a:

cubic binomial

Add or subtract this polynomial:

(3x²+5x-2)+(-2x²-4x+5)

In this case we should add the polynomials

(3x²+5x-2)+(-2x²-4x+5) =

Since there is nothing to do inside each parenthese, and there is nothing to distribute the parentheses fall off. Leaving:

3x²+5x-2-2x²-4x+5

Now combine like terms

(terms with the same base and power)

3x²-2x²+5x-4x-2+5=

x²+x+3

What are like terms?

Like terms are constants or variables with the same base raised to the same power. When this occurs in an addition or subtraction problem, we add or subtract their coefficients.

Examples:

• x+x = 2x
• ab³-5ab³ = -4ab³
• xyz⁴-xyz = xyz⁴-xyz 
  • (There are no like terms in this example therefore we cannot combine them.)

Add or subtract this polynomial:

(3x²+5x-2)-(-5x²+2x-2)

(3x²+5x-2)-(-5x²+2x-2)=

Distribute the -1 into the second parentheses

3x²+5x-2+5x²-2x+2=

Now combine like terms

3x²+5x²+5x-2x+2-2=

8x²+3x

Like terms are constants or variables with the same base raised to the same power. When this occurs in an addition or subtraction problem, we add or subtract their coefficients.

Examples:

• x+x = 2x
• ab³-5ab³ = -4ab³
• xyz⁴-xyz = xyz⁴-xyz 
  • (There are no like terms in this example therefore we cannot combine them.)

Multiply

(5x²)(4x³)

Group factors with like bases together then multiply:

(5x²)(4x³)

=(5·4)(x²·x³)

=20x⁵

Multiply

3xy²(3x-4y)

We will use the distributive property:

3xy²(3x-4y)

=3xy²·3x - 3xy²·4y

Group like bases together

3·3·x·x·y² - 3·4·x·y²·y

Now multiply

9x²y² - 12xy³

Can we go further? No.

Since these are not like terms!

Multiply

(x+3)(x-2)

We will use the distributive property

or FOIL to solve this one!

1. Multiply the First terms
   • (x+3)(x-2)      x · x = x²


2. Multiply the Outer terms
   
   • (x+3)(x-2)      x · -2 = -2x


3. Multiply the Inner terms
   • (x+3)(x-2)      3 · x -2 = 3x


4. Multiply the Last terms
   • (x+3)(x-2)      3 · -2= -6

(x+3)(x-2)= x² -2x +3x -6

Combine like terms

(x+3)(x-2)= x²+x-6

Simplify

5⁰

5⁰ = 5¹÷5¹ = 1

or

5⁰=1

Simplify

6^-3

6^-3 = 1 ÷ 6^-3

= 1 ÷ 216

Simplify

6y^-3

6y^-3 = 6 ÷ y³

Note: The 6 stays in the numerator because its

power is positive 6¹

Simplify

3w^-5 ÷ x^-6

3w^-5 ÷ x^-6 =

3x⁶ ÷ w⁵

Note: The 3 stays in the numerator because its

power is positive 3¹

Simplify

x^-8 ÷ 3y¹²

x^-8 ÷ 3y¹²

= 1÷ 3x⁸y¹²

Note: The 3 stays in the denominator because its

power is positive 3¹, y also stays in the denominator

because its power is positive y¹².

Multiplying by powers of 10

If the exponent is a positive integer,

move the decimal point to the ______.

Move the decimal point to the RIGHT!

125 x 10⁵ = 12,500,000

Multiplying by powers of 10

If the exponent is a negative integer,

move the decimal point to the ______.

Move the decimal point to the LEFT!

36.2 x 10^-3 = 0.0362

Find the value of this power of 10

10⁶

Since the integer is positve we move the decimal point

6 places to the RIGHT.

1,000,000

Find the value of this power of 10

10^-3

Since the integer is positve we move the decimal point

3 places to the LEFT.

0.001

Find the value of this expression

97.84 x 10⁴

97.84 x 10⁴ = 978,400

Find the value of this expression

19.2 x 10^-6

19.2 x 10^-6 = 0.0000192

Simplify

(7⁴)³

(7⁴)³ = 7⁴·7⁴·7⁴ = 7¹²

With multiplication of like bases we add exponents

Or we can use the Power of a Power property.

If a is any nonzero real number

and m and n are integers, then:

(a^m)ⁿ = a^m·n

(7⁴)³ = 7^4·3= 7¹²

This is way more efficient ;p

Simplify

(-5x^-2)^-3

(-5x^-2)^-3

We can use the Power of a Power property here.

If a is any nonzero real number

and m and n are integers, then:

(a^m)ⁿ = a^m·n

(-5x^-2)^-3 = (-5)^-3 ·x^{-2· -3}

= x⁶ ÷ (-5)³

= x⁶ ÷ (-5)³

= x⁶ ÷ -125

Simplify

3⁵ ÷ 3^-6

3⁵ ÷ 3^-6

Quotient of Powers Property

If a is any nonzero real number

and m and n are integers, then:

a^m ÷ aⁿ = a^m-n

3⁵ ÷ 3⁵ = 3^5-(-6) = 3¹¹