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5.1A Polynomials: Basics, Schemes and Mind Maps of Pre-Calculus

Stanford University Pre-Calculus

Method 2: FOIL. FOIL is an acronym for “First-Outer-Inner-Last” Consider the following example: 3. Page 4.

Typology: Schemes and Mind Maps

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5.1A Polynomials: Basics

A. Definition of a Polynomial

Apolynomial is a combination of terms containing numbers and variables raised to positive

(or zero) whole number powers.

Examples of Polynomials

             

      

NOT polynomials

 



 

(power is a fraction)

      

(power is negative)

B. Terminology

1. Degree

a. Term Degree: sum of powers in a term

   

the degree is



   

the degree is



the degree is



Discover Schemes and Mind Maps of Pre-Calculus Stanford University

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5.1A Polynomials: Basics

A. Definition of a Polynomial

A(or zero) whole number powers. polynomial is a combination of terms containing numbers and variables raised to positive Examples of Polynomials NOT polynomials (^) (power is a fraction) (^) (power is negative)

B. Terminology

Degree a. Term Degree: sum of powers in a term (^) the degree is (^) the degree is (^) the degree is

b. Polynomial Degree: maximum (not total) term degree (^) the degree is (^) the degree is

Descending Order We often write polynomials in order from the highest term degree to the the lowest. For instance, we rewrite ^ ^ ^ ^ ^ ^ as ^ ^ ^ ^ ^

C. Adding/Subtracting Polynomials

We combine like terms as before. Beware : minus signs and parentheses

1. Find ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

2. Find ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Find ^ ^ ^ ^ ^ ^ ^ ^ using FOIL

F irst: ^ ^ ^ ^ ^ ^

O uter: ^ ^ ^ ^ ^ ^ ^ ^

I nner: ^ ^ ^ ^ ^ ^ ^

L ast: ^ ^ ^ ^ ^ ^ ^

Thus we get ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

F. Multiplying Polynomials of Any Size

Method 1 : Distributive Property

If the problem is to expand to the terms of the second polynomial: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ , we distribute the

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Now use the distributive property again

Thus, after combining like terms, we get ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

A shortcut to the above method is called the factor table Method 2 : Factor Table You make a “tic-tac-toe” grid, and fill in the boxes with the products.

Consider ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Make factor table:

Then fill in the table with the products:

Collecting like terms:

5.1A Polynomials: Basics, Schemes and Mind Maps of Pre-Calculus

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1. Find ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

2. Find ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Find ^ ^ ^ ^ ^ ^ ^ ^ using FOIL

F irst: ^ ^ ^ ^ ^ ^ 

O uter: ^ ^ ^ ^ ^ ^ ^ ^

I nner: ^ ^ ^ ^ ^ ^ ^

L ast: ^ ^ ^ ^ ^ ^ ^

Thus we get ^  ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^  ^ ^ ^ ^ ^

If the problem is to expand     to the terms of the second polynomial: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ , we distribute the

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Thus, after combining like terms, we get ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^  ^ ^

Consider ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

1. Find ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

2. Find ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Find ^ ^ ^ ^ ^ ^ ^ ^ using FOIL

F irst: ^ ^ ^ ^ ^ ^

O uter: ^ ^ ^ ^ ^ ^ ^ ^

L ast: ^ ^ ^ ^ ^ ^ ^

Thus we get ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

If the problem is to expand to the terms of the second polynomial: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ , we distribute the

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Thus, after combining like terms, we get ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Consider ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^