an introduction to polynomials copyright scott storla 2015

Copyright Scott Storla 2015

An Introduction to Polynomials


Some Vocabulary for Polynomials


3 5x

Coefficient

Variable term Constant term


Notice a polynomial is a sum. You should think of

23 4x x as

2 13 1 4x x

Definition – Polynomial A polynomial in x is a single term, or a sum of terms, where each term is a variable term or a constant. Every variable term has a coefficient, the variable x, and an exponent of x that is a natural number.

Example: 32 3 5x x


One term 3 monomialk

Special names for the number of terms.

Two terms 3 7 binomialk

Three terms 3 7 trinomialk n


11 14n

14n

1. Write the polynomial as a sum with all coefficients and exponents explicit.

2. Discuss the polynomial in both general and specific terms.


1. Write the polynomial as a sum with all coefficients explicit.

2. Discuss the polynomial in both general and specific terms.

11 5y

5y


34 28 7 5 1a a a

The degree of a term

The degree of the entire polynomial is the same as the degree of the term with the largest exponent.

The degree of a polynomial

For each variable term use the exponent to decide on the degree of the term.

34 28 7 5 1a a a


Standard Form

The terms of the polynomial are written in decreasing order of degree from left to right.

3 25 7 2 Not in standard formb b b 3 27b 2 5 Standard formb b

3 21 5 7 2b b b 3 27 2 1 5b b b

To write a polynomial in standard form we imagine all operations are addition and all coefficients are explicit, then we use the commutative property to rearrange the terms, last we rewrite all explicit coefficients implicitly.

3 25 7 2b b b

3 27b 2 5b b


Standard Form

In practice people rearrange the terms of a polynomial “in their head”.

4 2 32 15 5x x x x

Write each polynomial in standard form.

4 3 25 2 15x x x x

7 3 9 29 12 15y y y y y

9 7 3 29 15 12y y y y y

5 7 3 8 2 47 5 8 3 6 4 2k k k k k k

8 7 5 4 3 23 5 7 2 8 4 6k k k k k k


1. Write the polynomial in standard form

2. Discuss the polynomial in general terms.

3. Discuss the polynomial term by term.

23 2x x 22 3x x

5 6 26 5 4 5 6n n n n 6 5 25 6 6 4 5n n n n

315 15y315 15y





5 7 3 8 2 47 5 8 3 6 4 2k k k k k k 8 7 5 4 3 23 5 7 2 8 4 6k k k k k k


Multivariable or “mixed” terms

With multivariable terms the degree of the term is the sum of the individual exponents. We don’t actually add the exponents.

A second degree termab42 A fifth degree termxy

2 3 23 z A seventh degree termx y


Multivariable or “mixed” terms

2 2 2a is often rewritten but is left alone.b ab a b

2 2 2 22 2 is usually rewritten 2 2y x x y

2

2

Even though 7 and 5 are both second degree terms, they are usually written in

the order 5 7 .

xy x

x xy

For standard form, terms of equal degree can be written in any order but often decisions are made using alphabetical order.





2 2 2 23 2xy x y x y 2 2 2 22 3x y x y xy

2 412 2 2 15j k jk k 4 215 12 2 2k j k jk

2 3 2 37 2 5ab a a b b 3 2 2 32 5 7a a b ab b


Some Vocabulary for Polynomials


Adding and Subtracting Polynomials


Terms are like if, in general, they’re counting the same sized unit.

Only like terms can be added or subtracted.


Like Polynomial Terms

6 , 11 Are like termsy y

26 , 11 Are liken ot t r s e mk k

5 56 , 11 Are like termsc c

Polynomial terms in one variable are like if the variable has the same exponent. Constant terms are also considered like.


Decide on the like terms

35 7t t t

2 2 24 2 7x x x x

3 2 3112 8 72y y y y

3 25 4 1k k k


32x

2 3 3 24 3 4 2x x x x

22x 4

Simplify


Simplify

4 3 4 2x x

5x 1

6 9 4 2 5y y y

0y 33

3 3 4 11 7 8x x x x

x 0x


3 2 2 36 9 4 2 5y y y y

Simplify

2 24 3 4 2x x

7 4 4 7 7 45 7 4 4 3p p p p p p

3 2 2 3 2 33 3 4 11 7 8x x x x x x x

2 3 2 32 5 4 9y y y y y

3 23 7y y

25 1x

46p

x

3 24 6y y y


2 15 4 2 11x xy xy x

Simplify

2 23 4 8x y yx yx xy

2 2 2 2 2 26 2x y y x xy x y

2 2 2 2 2 2 2 2 24 4 7 7 15 5i j i j j i i j ji i j j i

2 2 2 2 2 2 23 5 6 3 9 14a b ba a b b b a

3 9 13xy x

22 4x y xy

2 2 2 24x y x y xy

2 2 22i j ij

2 2 2 220 2 9 3a b a b b


Adding and Subtracting Polynomials

an introduction to polynomials copyright scott storla 2015

Documents