Math 9

5.1 Modeling Polynomials – Using Algebra Tiles

• We can use shapes or ‘tiles’ to represent various algebraic ‘polynomials’, and certain tiles are matched to certain ‘terms’.

• We have to start by defining and naming the various ‘terms’ that we will be working with.

What is a ‘term’?

• A term is any number or variable or both that can be separated from another term by a positive or negative sign. Eg. -7 or +2x or x2

• A term without any variables is called a constant. Eg. -12 or +25 or +2.5

• ‘x’ can be anything but ‘4’ will always be ‘4’…it’s constantly ‘4’

• The rest of our terms will then need to be described too…

What is a ‘term’?

• A term is any number or variable or both that can be separated from another term by a positive or negative sign. Eg. -7 or +2x or x2

• But what are the parts called?

-3x2the sign

the coefficient the variable(s)

the exponent

Coefficient – the numerical factor of a term.

x3

25x

Coefficient

3 and –5 are the numerical coefficients of these 2 terms.

What is a ‘term’?

• What is the ‘sign’, ‘coefficient’, ‘variable’ and ‘exponent’ of the following terms?

• -25a4

• 7.4b3

What is a ‘term’ vs an ‘polynomial’?

• A term is any number or variable or both that can be separated from another term by a positive or negative sign. A term can contain more than one variable. Eg. -7 or +2xy or x2

• A polynomial is made up of ‘terms’ separated by positive or negative signs. Eg.

• x2 +5x -7 (3 terms)• 2a +3 (2 terms)• -13 (1 term)

How do we name polynomials?

• If a polynomial has one term in it, it is called a monomial. Eg. +15 or -5ab or x2

• If a polynomial has two terms in it, it is called a binomial. Eg. 6a -3 or 3x2 +8

• If a polynomial has three terms in it, it is called a trinomial. Eg. -4d2 +10d -7

•MONOmial- has ONE term Ex. 3x2 4x

½ bc 7x2y

Ex. 3x2 4x

½ bc 7x2y

•BInomial- TWO terms

Ex. 3x + 72y2 - y

½ z3 - 4zy

Ex. 3x + 72y2 - y

½ z3 - 4zy

•TRInomial – THREE terms Ex. 3x2 + 7x - 61 - 4x - 3y2

Ex. 3x2 + 7x - 61 - 4x - 3y2

What do you think an expression with 3 terms would be

called?

What do you think an expression with 3 terms would be

called?

An algebraic expression that contains a term with a variable in the denominator, such as

or the square root of a variable such as is NOT a polynomial.

n

n

3

How do we name polynomials?

• So what would each of these be called?

• (3f -7)• (2xyz2)• (-6.7x)• (12a -3b +7c)

How do we name polynomials?

• The degree of a polynomial is determined by the term with the largest exponent.

• We prefer to write expressions in order.

• So we would write (-a3 +3a +9 –4a6 +10a2) as:• -4a6 –a3 +102 +3a +9

descending

Using algebra tiles

• We can start out by using tiles to represent constants.

• A tile represents +1.• A tile represents -1.• The reason for this is that the dimensions of

the tile are 1 by 1 = 1. (zoomed in)

1

1

Using algebra tiles

• So we can represent the following constants with the following tiles:

• +6

• -3

Using algebra tiles

• Now on to the variables:• A tile represents +x.• A tile represents –x.• The reason for this is that the dimensions of

the tile are 1 by x = 1x. (zoomed in)

x

1

Using algebra tiles

• So we can represent the following terms with the following tiles:

• +6x

• -3a

Using algebra tiles

• A tile represents +x2.

• A tile represents –x2.

• The reason for this is that the dimensions of the tile are x by x = x2.

x

x

Using algebra tiles

• So we can represent the following terms with the following tiles:

• +3x2

• -3a2

Write the expression represented by these tiles:

3x2-2x 5

= 3x2 – 2x + 5

Represent each polynomial using algebra tiles…

a) –3x2 + 1 b) 6 – x

c) 3x2 – x – 4 d) 5

Assignment

•Page 214: 5 - 9, 12, 13, 15, 16, then 11,

Classify the following as monomials, binomials, trinomials or polynomials:

51 2x x

xyx 42

1 2 23 x25

2

y

yx

yx 4

32

22 x

y255

1a

Polynomials:

A polynomial is one term or the sum of terms whose variables have whole number exponents.

• A term is a number, a variable, or the product of numbers and variables.Terms are separated by signs (+, -).

• A Constant is a number whose value doesn’t change.

• If there are 2 they represent 2x.

• If there is a , it represents -1.

• Together, they would represent the expression 2x-1

• Some expressions contain terms such as “x2”.

• We use a large square to represent these.

These can also be negative as well.

-x2

x2