an introduction to polynomials copyright scott storla 2015
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Copyright Scott Storla 2015
An Introduction to Polynomials
Copyright Scott Storla 2015
Some Vocabulary for Polynomials
Copyright Scott Storla 2015
3 5x
Coefficient
Variable term Constant term
Copyright Scott Storla 2015
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
Copyright Scott Storla 2015
One term 3 monomialk
Special names for the number of terms.
Two terms 3 7 binomialk
Three terms 3 7 trinomialk n
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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.
Copyright Scott Storla 2015
1. Write the polynomial as a sum with all coefficients explicit.
2. Discuss the polynomial in both general and specific terms.
11 5y
5y
Copyright Scott Storla 2015
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
Copyright Scott Storla 2015
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
Copyright Scott Storla 2015
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
Copyright Scott Storla 2015
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
Copyright Scott Storla 2015
1. Write the polynomial in standard form
2. Discuss the polynomial in general terms.
3. Discuss the polynomial term by term.
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
Copyright Scott Storla 2015
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
Copyright Scott Storla 2015
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.
Copyright Scott Storla 2015
1. Write the polynomial in standard form
2. Discuss the polynomial in general terms.
3. Discuss the polynomial term by term.
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
Copyright Scott Storla 2015
Some Vocabulary for Polynomials
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Adding and Subtracting Polynomials
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Terms are like if, in general, they’re counting the same sized unit.
Only like terms can be added or subtracted.
Copyright Scott Storla 2015
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.
Copyright Scott Storla 2015
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
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32x
2 3 3 24 3 4 2x x x x
22x 4
Simplify
Copyright Scott Storla 2015
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
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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
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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
Copyright Scott Storla 2015
Adding and Subtracting Polynomials
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