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 x 2 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 x 2 But what are the parts called? -3x2-3x2 the sign the coefficient the variable(s) the exponent

Coefficient – the numerical factor of a term. 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? -25a 4 7.4b 3

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 x 2 A polynomial is made up of ‘terms’ separated by positive or negative signs. Eg. x 2 +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 x 2 If a polynomial has two terms in it, it is called a binomial. Eg. 6a -3 or 3x 2 +8 If a polynomial has three terms in it, it is called a trinomial. Eg. -4d 2 +10d -7

MONOmial- has ONE term Ex.3x 2 4x ½ bc 7x 2 y Ex.3x 2 4x ½ bc 7x 2 y BInomial- TWO terms Ex.3x + 7 2y 2 - y ½ z 3 - 4zy Ex.3x + 7 2y 2 - y ½ z 3 - 4zy

TRInomial – THREE terms Ex.3x 2 + 7x x - 3y 2 Ex.3x 2 + 7x x - 3y 2 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.

How do we name polynomials? So what would each of these be called? (3f -7) (2xyz 2 ) (-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 (-a 3 +3a +9 –4a 6 +10a 2 ) as: -4a 6 –a a +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 +x 2. A tile represents –x 2. The reason for this is that the dimensions of the tile are x by x = x 2. x x

Using algebra tiles So we can represent the following terms with the following tiles: +3x 2 -3a 2

Write the expression represented by these tiles: 3x23x2 -2x 5 = 3x 2 – 2x + 5

Represent each polynomial using algebra tiles… a) –3x 2 + 1b) 6 – x c) 3x 2 – x – 4d) 5

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

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

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 “x 2 ”. We use a large square to represent these. These can also be negative as well. -x 2 x2x2