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
• 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
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