1. Polynomials
Unit 5

2. Modeling Polynomials Polynomial:
 An algebraic expression that contains one term or a sum of terms.
The term(s) may contain variables (which will have whole number exponents).
 And a term may be a number.

3. Modeling Polynomials 3x + 1
Is a polynomial. It contains a variable (whose exponent is 1) and numbers
Since it is an expression, there is no equal sign

4. Modeling Polynomials 3x + 1 It has 2 terms
A term is a number, or a variable, or the product of numbers and variables.
Terms are separated by a + or a –
Therefore 3x is one term and 1 is another term

5. Modeling Polynomials 3x + 1
In the term 3x the 3 is the numerical coefficient.
This is the number in front of the variable
It is the numerical factor of a term
X is called the variable
1 is called the constant term
There is no variable attached to this number

6. Modeling Polynomials
We can classify a polynomial by the number of terms it has.
Polynomials with 1, 2 or 3 terms have special names.

7. Modeling Polynomials Monomial has 1 term Binomial has 2 terms
5x or 9 or -2p2
Binomial has 2 terms
2c-5 or 2m2 + 3m or x + y
Trinomial has 3 terms
2h2-6h+4 or x+y+z

9. Modeling Polynomials
Just as numbers are the building block of arithmetic
Polynomials are the building blocks of algebra!
We use algebra tiles the same as base ten blocks but now we are representing polynomials

10. Try Some
Use the algebra tiles provided
Model each of the following:

11. 5.1 – Modeling Polynomials
Write an equation for each model. Is it a monomial, binomial or trinomial? A B

12. 5.1 – Modeling Polynomials
Use algebra tiles to model this polynomial:
4 – 2x2 + x
What is the variable?
How many terms are there?
What are the coefficients?
Are there any constants?

13. 5.1 – Modeling Polynomials
When is a number or expression NOT a polynomial?
When a variable is in the denominator
Square root of a variable
X-2 X1/2 3/x
A polynomial is one term (or the sum of terms) whose variables have whole-numbered exponents

14. 5.1 – Modeling Polynomials
Each part of a polynomial expression is called a term
Terms begin with a + or a – (unless they are the first term)
2x2 + 4x + 3x + 2 – 8 ____ terms
Simplify ____ terms

15. 5.1 – Modeling Polynomials
We use monomial, binomial and trinomial to describe how many terms a simplified expression has
Simplify where you can and classify each polynomial

16. 5.1 – Modeling Polynomials
For each term, the number that is multiplied by the variable is called the coefficient
If there is no variable, the terms is called a constant A
What are the coefficients and constants for each expression? B C

17. 5.1 – Modeling Polynomials
The degree of a term is determined by the exponent of the variable
If there is no variable it has a degree of 0
The term 2x2 has a degree of 2
The term 2x has a degree of 1
The term 2 has a degree of 0

18. 5.1 – Modeling Polynomials
In your notebook write down the polynomials and identify their degree:
x2 4x 3x x x12

19. 5.1 – Modeling Polynomials
Just like polynomials have names based on their number of terms they are also named based on their degree.
Degree
Name
Constant 1
Linear 2
Quadratic 3
Cubic 4
Quartic 5
Quintic
N >6
nth degree

20. 5.1 – Modeling Polynomials
In your notebook write down the polynomials and identify their names:
x2 4x 3x x x12

21. 5.1 – Modeling Polynomials
The term with the greatest exponent determines the degree of the polynomial
2x3 + 4x2 – 3x + 8 is a _____ degree polynomial

22. 5.1 – Modeling Polynomials
Identify the degree of the following polynomials
x2 + x – 3 -2x2 – 3 2x3 + 3x
-2x4 -3x x x2 + x5 + 2x - 2

23. 5.1 – Modeling Polynomials
Polynomials should always be written from highest degree to lowest degree term
Rearrange the following polynomials so that they are in order:
X + 5 – x x2 – 3 + 4x 8 – 2x4 – 3x + x2

24. 5.1 – Modeling Polynomials
Math Practice – remember THIS is where your quiz and test questions come from
Page 214
Questions – 4ace, 5, 6, 7, 8, 9ace, 11ace, 12, 13 ace, 14 and 18

25. Homework Review Put your name on the sticky note
Write a polynomial that fits ALL the following criteria:
3 terms
Complete polynomial has a degree of 2
One term has a degree of 1
Includes a constant term of 12
One negative term
Two different coefficients

26. 5.2 Like & Unlike Terms
Any two opposite colored tiles of the same size has a sum of zero – these tiles are like terms
Like Terms – terms that have the same variable, raised to the same exponent

27. 5.2 Like & Unlike Terms

28. 5.2 Like & Unlike Terms
For the sake of visual clarity green tiles will be positive and red will be negative

29. 5.2 Like & Unlike Terms

30. 5.2 Like & Unlike Terms
-x2 +3x - 4

31. 5.2 Like & Unlike Terms

32. 5.2 Like & Unlike Terms

33. 5.2 Like & Unlike Terms 8x

34. 5.2 Like & Unlike Terms What would be the perimeter of these shapes? A
Will have a polynomial
A – 6x +4
B – 6x+2

35. 5.2 Like & Unlike Terms Simplifying a polynomial with two variables:
Group like terms
Combine (add/subtract) like terms
Simplify the following:
A - 11 – 5x2 + 3y + 4x – 5y +8y2 – 2x – 6 – 5y2 – 2x2
B - 11x – 2y2 – 3x2 + 4 – y + 10 – 4x – 3y – 5y – x

36. 5.2 Like & Unlike Terms
Math Practice – remember THIS is where your quiz and test questions come from
Page 222
Questions 4, 5, 6, 7, 8ace, 11ace, 12ace, 13ace, 14ace, 17, 18, 19ac, 22

37. 5.3 Adding Polynomials To add polynomials you collect like terms
Three methods:
Combine algebra tiles, group like tiles and cancel them out
Add horizontally
Add vertically

38. 5.3 Adding Polynomials

39. 5.3 Adding Polynomials

40. 5.3 Adding Polynomials

41. 5.3 Adding Polynomials

42. 5.3 Adding Polynomials

43. 5.3 Adding Polynomials
Write a polynomial for the perimeter of this rectangle.

44. 5.3 Adding Polynomials

45. 5.3 Adding Polynomials Math Practice Page 228
Questions – 3, 5, 6, 8 (pick 2), 9 (pick 2), 10 (this question will for sure appear on a future quiz), 12

46. 5.4 Subtracting Polynomials
Just like adding polynomials, you collect the like terms
There is only 1 MAJOR difference: when you drop the brackets you change the sign of each term after the subtraction sign

47. 5.4 Subtracting Polynomials
You must get the opposite of EVERY term in a polynomial.

48. 5.4 Subtracting Polynomials
When subtracting polynomials you must remember to ADD the OPPOSITE of every term.

49. 5.4 Subtracting Polynomials
How would you demonstrate this with algebra tiles?
Model the following using tiles
(-2x2 + x – 11) – (x2 – 3x + 2)

50. 5.4 Subtracting Polynomials
=y2 –8y+7

51. 5.4 Subtracting Polynomials
Math Practice
Page – 234 – 236
Questions – 4, 7 (pick three), 8 (pick three), 9, 10 and 13

52. 5.5 Multiplying and Dividing a Polynomial by a Constant
We will only be multiplying and dividing polynomials by MONOMIALS
Constants
Contain 1 variable
You will be expected to know symbolically, using area models and algebraically.

53. 5.5 Multiplying and Dividing a Polynomial by a Constant

54. 5.5 Multiplying and Dividing a Polynomial by a Constant

55. 5.5 Multiplying and Dividing a Polynomial by a Constant

56. 5.5 Multiplying and Dividing a Polynomial by a Constant

57. 5.5 Multiplying and Dividing a Polynomial by a Constant
Dividing Symbolically

58. 5.5 Multiplying and Dividing a Polynomial by a Constant

59. 5.5 Multiplying and Dividing a Polynomial by a Constant

60. 5.5 Multiplying and Dividing a Polynomial by a Constant

61. 5.5 Multiplying and Dividing a Polynomial by a Constant
Math Practice
Page 246 – 248
Questions – 3d, 4d, 5a, 9b, 10b, 11ace, 12, 13ace, 22d, 23d