Unit 5: Polynomials
5.1. Modelling Polynomials
A constant is a fixed quantity that does not change. Examples: 3, -6, π, 1/2
A placeholder is any symbol that replaces a number.
Example: In the equation, 3 + = 7, the placeholder is the .
A variable is a placeholder in the form of a letter
Note: The most commonly used letter is , but any letter can be used. Lower case letters are generally used in algebra.
A term can be: a constant,
the product any number of variables, or
the product of a constant and any number of variables.
Terms are generally written in simplified form with any letters in alphabetical form and using powers where applicable.
Note: It is very important to understand that the only operation in a term is multiplication.
Some examples of terms are the following:
i) –5 (an integer/real number) ii) (a variable)
iii) (a rational/real number)
iv) (since is the same as xy, thus multiplication)
v) (since this is the product of a real number and variables) vi) (this really means , which is a product)
The Numerical Coefficient is the real number contained in a term.
The Literal Coefficient is(are) the variable(s) contained in a term. Examples:
Term Numerical Coefficient
Literal Coefficient 5
3 1 -1
A term also has a degree. It is the sum of the exponents of the literal coefficient in the term only.
Examples: Give the degree of the following terms.
a) degree = 1 (remember that is the same as )
b) degree = 2 (sum is 1 + 1)
c) degree = 2
d) –2 degree = 0 (-2 is the same as )
Note: terms do not contain quotients or square roots of variables. The following are NOT terms:
i) (this means , which is NOT a product)
real number, a variable, or the sum of any number of terms.
Example: is a polynomial expression which contains 3 terms:
- the term,
- the variable term, and 1 - the constant term.
Also, note that
A monomial is a polynomial containing only one term. A term is also a monomial.
i.e. , , or
For non-examples, see non-examples of terms above.
A binomial is a polynomial containing two terms, separated by addition.
i.e. , , or
Non-examples of binomials: (no sum), (no sum), ( is not a monomial)
A trinomial is a polynomial containing three terms, separated by addition.
Non-examples of trinomials: (division by variable), (no sum)
Recall Algebra Tiles from previous years
Recall that when using coloured tiles:
yellow is positive
red is negative
When using shaded and un-shaded tiles a key is necessary. Key: un-shaded is positive
shaded is negative
1. What does each of the individual tiles represent?
1 tile x tile x2 tile
-1 tile -x tile -x2 tile
2. What does each of the following sets of tiles represent?
b) x + (-2) or x - 2
Note that we are using x but we really could use the tiles to represent any variable. Zero Pairs: Equivalent positive and negative tiles cancel each other out!
1. +1 + (-1) = 0
2. +x + (-x) = 0
Other Tiles to note:
When we have only one variable, the length of the tile does not matter. However, when we are using more than one variable we need tiles of different lengths.
The tile at the right is an tile.
The tile below has a different length. What might we call it?
Remember the tile looks like this: and the y-tile looks like this:
Keeping in mind that tiles are named for their area, what do you suppose the tile below might be called?
x times y = xy tile
Note too, the -y tile
Model for students and have them record it in their notes
Have students model the expressions in the investigate p. 210 and sketch the results.
Polynomials are usually written in descending order – terms from highest degree to lowest degree. However, equivalent polynomials can be written by rearranging the terms. Note that the sign of the terms has to remain with the term. As long as the expression can be represented using the same number of each type of tile the expressions are equivalent.
Example: is equivalent to
Go through examples 1-3 pp. 212-213
Discuss p. 213 #1 & 2
Set pp. 184-185 in practice book for P3 student only
5.2. Like Terms & Unlike Terms
Expressions using congruent tiles (same shape and size) represent like
Likewise, in the expression: , and are unlike terms because they require the use of different tiles.
Without the use of tiles, we can tell if two terms are like if they have the same literal coefficient.
See bottom p. 218
Example. Which of the following pairs of terms are like terms? Which are unlike terms? Explain.
(a) , Like terms – literal coefficient for both (b) , Unlike terms – different exponents for
(c) , Unlike terms – different variables
(d) , Like terms – literal coefficient for both
In an expression, we usually combine like terms.
See example in connect top p. 218
Note: An expression is simplified when:
We use the fewest number of algebra tiles possible. There are no like terms and no coefficients of zero. See example top p. 219 using tiles
Also, recall equivalent polynomials.
See examples 1-4 pp. 219-221 using overhead tiles where necessary.
Note the use of different variables.
Complete p. 222#4 & 5 on Smartboard
5.3. Adding Polynomials
Like numbers, polynomial expressions can be added to each other.
To add polynomials we add like terms.
See tile method in connect p. 226
Additional Example: Add using algebra tiles
To add polynomials without tiles we group the like terms and then simplify.
Example: = = Think:
See examples 1-3 pp. 227-228
Column method for addition in example 1 Application of perimeter in example 2
Using more than one variable in example 3
Complete pp. 228-229 #3, 4, 5, 6 together on Smartboard
5.4. Subtracting Polynomials
See tile method in connect p. 232 using tiles
As with numbers, to subtract polynomial expressions we “add the opposite” of the second expression.
Step 1. Begin with the tiles that represent
Step 2. Add the tiles that represent the opposite of , that is
. Note that it is important that we add the opposite of the entire second expression, not just the first term.
Step 5. Give the simplified answer :
To subtract polynomials without tiles we use the properties of integers.
See example connect bottom p. 232
Note that we must “add the opposite” of EVERY term in the second expression.
Additional Example: =
The vertical method is VERY useful here in reducing the amount of work and reducing the change of mistakes.
Note that we can check by addition
Discuss p. 234 #3
We can subtract to find a missing side in a shape if we know the perimeter and the other sides.
Example: The length two sides of a triangle are given by the
expressions and . The perimeter is given by expression . What is the expression for the length of the third side of the triangle?
length = Perimeter – length of each known side =
= = =
In a rectangle, we have to be careful.
Example: The length of a rectangle is given by the expression . The perimeter is given by expression . What is the
expression for the width of the rectangle?
2 widths = Perimeter – 2 lengths = = = = = Width =
Complete p. 234 #4 & 5d together
5.5. Multiplying and Dividing a Polynomial by a Constant
Multiplying by a Scalar
Repeated addition and/or the area model is very useful here
See connect p. 242
See example 1 method 1 p. 243
The distributive model is also useful when multiplying a polynomial by a constant
See example 1 method 2 p. 243
Additional examples: 1.)
Think 2 set of
Or use the distributive property/area model
Think 3 set of
Or use the distributive property/area model
The area model is not very useful when the scalar is negative.
Dividing by a Scalar
See p. 244
Also see example 2 p. 245
Discuss p. 246 #2 and the usefulness of method 2
5.6. Multiplying and Dividing a Polynomial by a Monomial
Multiplying by a monomial
Note: We cannot use the repeated addition model here.
We can recall our previous work with exponents or the area model and/or distributive property can be very useful here.
See connect pp. 250-251 and example 1 pp. 251-252
Complete p. 255 #4, 6, 7 together on Smartboard Set pp. 256-257 #11ace, 12-14, 19, 20
Dividing by a monomial
See pp. 252-253 and example 2 p. 254
Discuss #3 p. 254
Additional example: Given four x2-tiles and eight x-tiles can you
create rectangle with one dimension 4x? What is the other dimension?
Solution: the missing length is x + 2
Complete p. 255 #5, 8 together on Smartboard