Wize High School Grade 9 Math Textbook > Polynomial Expressions

Definitions/Terminology

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The Language of Algebra
In math we like to use symbols to help represent real-world problems.

Example
A small cup of coffee is $1.75. Let's say you want to buy coffee for you and all of your friends, and you want to calculate how much you will have to pay in total.

In English words:

You'll have to pay $1.75×the number of cups of coffee you want to buy\boxed{\$1.75\times{\colorFive{\bm{\text{the number of cups of coffee you want to buy}}}}}$1.75×the number of cups of coffee you want to buy​.

Using Symbols:

If we let x\bcfi xx be the number of cups of coffee you want to buy, then you'll have to pay $1.75x\boxed{\$1.75\bcfi x}$1.75x​

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The building blocks of the English Language→AlphabetsThe building blocks of the Algebra Language→Numbers  and  Variables\boxed{\begin{array}{ccc}
\text{The building blocks of the English Language} &\to& \bold{{Alphabets}}\\\\
\text{The building blocks of the Algebra Language} &\to& \colorFour{\bold{{Numbers~~and~~Variables}}}
\end{array}}The building blocks of the English LanguageThe building blocks of the Algebra Language​→→​AlphabetsNumbers  and  Variables​​


A number is used to represent a quantity or amount. If we have a number that is alone, we call it a constant.
Examples
333 can represent 3 cats, 3 meters, 3 grams, $3, etc.
−5-5−5 can represent −5°-5\degree−5°, owing $5, 5 steps left, etc.
0.150.150.15 can represent $0.15, 0.15 ml, 0.15 km, etc.



A variable is any symbol (usually letters) used to represent a quantity (number), this quantity can change and take on many different values!
Examples
xxx is the number of people on a certain bus
hhh is the height of a tree
www is the weight of an apple
lll is the length of a rectangle and www is the height of that rectangle

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In English, we can string a bunch of alphabets together→WordsIn Algebra, we can have a product of variables and constants→Terms\boxed{\begin{array}{ccc}
\text{In English, we can string a bunch of alphabets together} &\to& \bold{{Words}}\\\\
\text{In Algebra, we can have a product of variables and constants} &\to& \colorFour{\bold{{Terms}}}
\end{array}}In English, we can string a bunch of alphabets togetherIn Algebra, we can have a product of variables and constants​→→​WordsTerms​​
Examples 
5x5x5x
−0.5xy-0.5xy−0.5xy
x3x^3x3
l2w3l^2w^3l2w3
x6\dfrac{x}{6}6x​

Notes
Multiplication is usually shown as 3×y3\times y3×y or (3)(y)(3)(y)(3)(y), but when multiplying by a variable, we can just show it as 3y3y3y 
A number that is being multiplied by a variable has a special name called coefficient.

Watch Out!
Is a number or variable by itself a term?
Is −5-5−5 a term?    Yes!   
Is xxx a term?    Yes!   

What is the coefficient of xxx?
We can rewrite xxx as 1x1x1x, and see more clearly that the coefficient is 11
1!

Practice: The Language of Algebra
Fill in the blanks with the words variable, constant, coefficient.

−5 x  + 7\Large{\bcfi{-5}~\bcf{x}~~\bct{+~7}}−5 x  + 7
−5\bcfi{-5}−5 is called a 
x\bcf{x}x is called a 
7\bct{7}7 is called a 

I don't know

Practice: Algebra Definitions

Fill in the blanks.
3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b is called a 
The xxx in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b is called a 
The 555 in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b is called a 
The −4xy-4xy−4xy in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b is called a 
How many terms are there in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b? 
How many variables are there in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b? 
The 333 in the term 3x23x^23x2 is called the 
What is the coefficient of the second term in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b? 
What is the coefficient of the fourth term in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b? 
What is the coefficient of the fifth term in 3x2−4xy+5+a−b3x^2-4xy+5+a-b3x2−4xy+5+a−b? 

*If your answer is a negative number, don't forget to type the - sign (for example -7)

*If your answer is a positive number, you don't have to type the + sign (for example 10)

I don't know

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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

Wize High School Grade 9 Math Textbook > Polynomial Expressions

Definitions/Terminology

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