Wize High School Algebra I Textbook (Common Core) > Polynomial Expressions

Definitions/Terminology

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Polynomials

In English, we can put a bunch of words together→SentencesIn Algebra, we can add and subtract terms→Polynomials(Algebraic Expressions)\boxed{\begin{array}{ccc}
\text{In English, we can put a bunch of words together} &\to& \bold{{Sentences}}\\\\
\text{In Algebra, we can add and subtract terms} &\to& \colorFour{\bold{{Polynomials}}}\\
&&\text{(Algebraic Expressions)}
\end{array}}In English, we can put a bunch of words togetherIn Algebra, we can add and subtract terms​→→​SentencesPolynomials(Algebraic Expressions)​​

Examples
3+5x3+5x3+5x
5x−2x+35x-2x+35x−2x+3
3xy+x−13xy+x-13xy+x−1
2x2−x−3y+12x^2-x-3y+12x2−x−3y+1
x2−5\dfrac{x}{2}-52x​−5

Notes
Terms are separated by +++ and −-− signs.
A single constant, single variable, or any term by itself is also a polynomial!
You can think of this as adding 0 to the single constant, the single variable, or the term

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Watch Out!
A polynomial CANNOT have
division by a variable
ex. 2x\dfrac{2}{x}x2​,   xy\dfrac{x}{y}yx​

negative exponents
ex. x−3x^{-3}x−3,   x2y−2x^2y^{-2}x2y−2

decimal exponents
ex. x0.3x^{0.3}x0.3,   x2y1/2x^2y^{1/2}x2y1/2

an infinite number of terms
ex. 1+2+4+8+...1+2+4+8+...1+2+4+8+...

Practice: Polynomials
Select ALL of the expressions that are polynomials.

A) 3xy2−x+y+7.53xy^2-x+y+7.53xy2−x+y+7.5

B) x5+3y2−1\dfrac{x}{5}+\dfrac{3y}{2}-15x​+23y​−1

C) 3x−y2+1\dfrac{3}{x}-y^2+1x3​−y2+1

D) 2.5x3y7−122.5x^3y^7-\dfrac{1}{2}2.5x3y7−21​

E) 3x2y2.3+x−3y−63x^2y^{2.3}+x-3y-63x2y2.3+x−3y−6

I don't know

Practice: Polynomials
How many terms are in the following polynomials?

a) 3x−63x-63x−6

b) −7x2−x+1-7x^2-x+1−7x2−x+1

c) 555

d) 2xy−4a+7x−12xy-4a+7x-12xy−4a+7x−1

I don't know

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Monomials, Binomials, Trinomials
Polynomials that have 1, 2, or 3 terms have special names:
Monomial - a polynomial with only a single term
Examples - 4x,     −5,     −3xy24x,~~~~~-5,~~~~~-3xy^24x,     −5,     −3xy2
Binomial - a polynomial with exactly 2 terms
Examples - −2x+1,     5+7xy,     3x2+x-2x+1,~~~~~5+7xy,~~~~~3x^2+x−2x+1,     5+7xy,     3x2+x
Trinomial - a polynomial with exactly 3 terms
Examples - 3x2+2x−1,     −4xy+5−x3x^2+2x-1,~~~~~-4xy+5-x3x2+2x−1,     −4xy+5−x

Wize Tip
In math and science, 
"mono" means 1 (ex. monochromatic means 1 colour)
"bi" means 2 (ex. bicycle has 2 wheels)
"tri" means 3 (ex. triangle has 3 sides)

Practice: Monomial, Binomials, Trinomials

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Degree of a Polynomial
The degree of a term is the total of the exponents on all of its variables.

Examples
The term −5x4-5x^4−5x4 has degree    4   
The term xxx has degree    1   
The term 6x2y56x^2y^56x2y5 has degree    2 + 5 = 7   
The term −xy2z-xy^2z−xy2z has degree    1 + 2 + 1 = 4   


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The degree of a polynomial is the highest degree of all of its terms.

Examples
The degree of 3x4−2x2+13x^4-2x^2+13x4−2x2+1 is    4   
The degree of 5x2y−7xy5x^2y-7xy5x2y−7xy is    3   
The degree of −2x3+6y3-2x^3+6y^3−2x3+6y3 is    3   

Practice: Polynomials
What is the degree of the following polynomials?

a) 3x2−2x−13x^2-2x-13x2−2x−1

b) 5xy+x−65xy+x-65xy+x−6

c) 6−x3+x2y46-x^3+x^2y^46−x3+x2y4

I don't know

Extra Practice

Basics of Polynomials

Practice: Basics of Polynomials
Which of the following are polynomials?

Basics of Polynomials

Practice: Basics of Polynomials
Which of the following are polynomials?

Basics of Polynomials

Practice: Basics of Polynomials
Which of the following are polynomials?