1. Polynomials and Factoring
Chapter 8
Polynomials and Factoring

2. Definitions Binomial- a polynomial of two terms
Degree of polynomial- the greatest value of exponent of any term of the polynomial
Degree of monomial- the sum of the exponents of the variables of a monomial
Difference of two squares- a difference of two squares is an expression of the form a²-b². Factors to (a + b)(a - b)

3. Definitions
Factoring by grouping- a method of factoring that uses the distributive property to remove a common binomial factor of two pairs of terms
Monomial- a real number, a variable, or a product of a real number and one or more variables with whole-number exponents

4. Definitons
Perfect square trinomial- any trinomial of the form a² + 2ab + b² or a² - 2ab + b²
Polynomial- a monomial of the sum or difference of two or more monomials. A quotient with a variable in the denominator is not a polynomial
Standard form of a polynomial- the form of a polynomial that places the terms in descending order by degree
Trinomial- a polynomial of three terms

5. Adding and subtracting polynomials
8.1
Adding and subtracting polynomials

6. Essential Understanding
You can use monomials to form larger expressions called polynomials. Polynomials can be added or subtracted.

7. Degree of monomial
The degree of a monomial is the sum of the exponents of its variables only.
The degree of a nonzero constant is 0.
Zero means it has no degree

8. Finding the Degree of Monomial
Examples: 11 ⅞ 12 ∏ 92
Any polynomial whose largest term has an exponent that is not stated, and no variables has a degree of zero

9. Finding the Degree of Monomial
Examples: x z a y b p
Any polynomial whose largest term has a variable with an implicit exponent of one has a degree of one

10. Finding the Degree of Monomial
Examples:
Degree of 2
2xy
4z²
7pq
Degree of 3
xyz x³
Xy²
Degree of 4
x²y²
wxyz
Any mononomial whose largest term has an exponent or has multiple terms with exponents has the sum of the exponents as its degree

11. Combining Like Monomials
Example
Explanation
To find degree, sum exponents
Even if bases are not the same
x²x²
2+2=4
6x³y²
3+2=5
-7x⁴z²
4+2=6
uvwxyz =6
Combine exponents
Term has a degree of four
Term has a degree of five
Term has a degree of six

12. Standard Form of Polynomial
Examples in Standard Form
Examples NOT in Standard Form
In standard form, the exponents (degree) descend as the terms are listed.
5x² + 7x - 10
12x³ - 6x + 9
19x⁴ - 8x³ - 5x
x⁴ + x³ + x² + x + 1
An expression is not in standard form if the degrees do not descend in order
6x-12x²
1-3x⁴+2x+x³
x² + 12x⁴ + 12x

13. Degree of a Polynomial
The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent.
The degree of 3x⁴ + 5x² - 7x + 1 is four

14. Degree of Polynomial Polynomial Degree Name of Degree Number of Terms
Name using Number of Terms 10
None (0)
Constant
One
monomial x
One (1)
Linear
7x-5
Two
binomial
5x²
Two (2)
Quadratic
12x²+144
3x²+9x+12
Three
trinomial
8x³
Three (3)
Cubic
4x³+10x
x³+3x²+x+1
Four
polynomial
x⁴+x³+x²+x+1
Four (4)
Quartic
Five

15. Classifying Polynomials
To classify a polynomial
Ensure all like terms are combined
Find the term with the greatest exponent
List term with greatest exponent
Find term with next greatest exponent
List term with next greatest exponent
Continue process until you reach the constant term or, if no constant term exists, the term with the least greatest exponent

16. Multiplying and factoring
8.2
Multiplying and factoring

17. Essential Understanding
You can use the distributive property to multiply a monomial by a polynomial

18. Multiplying Polynomials

19. Greatest Common Factor

21. Multiplying binomials
8.3
Multiplying binomials

22. Essential Understanding
There are several ways to find the product of two binomials including models, algebra, and tables.

23. Multiplying Binomials

24. Multiplying Binomials Using the Distributive Property

25. Multiplying Binomials Using a Table

26. FOIL FOIL stands for first outer inner last
This method does not work for multiplying two polynomials with more than two terms for each

27. Multiplying special cases
8.4
Multiplying special cases

28. Essential Understanding
There are special rules you can use to simplify the square of a binomial or the product of a sum and difference.
Squares of binomials have two forms:
(a + b)²
(a – b)²

29. The Square of a Binomial
The square of a binomial is the square o the first term plus twice the product of the two terms plus the square of the last term.
(a + b)² = a² + 2ab + b²
(a – b)² = a² -2ab + b²

30. The Product of a sum and difference
The product of the sum and difference of the same two terms is the difference o their squares.
(a + b)(a – b) = a² - b²

31. 8.5
Factoring x²+bx+c

32. Essential Understanding
You can write some trinomials of the form x² + bx + c as the product of two binomials

33. Factoring x² + bx + c
Use a table to list the pairs of factors of the constant term c and the sums of those pairs of factors.

34. 8.6
Factoring ax²+bx+c

35. Essential Understanding
You can write some trinomials of the form ax² + bx + c as the product of two binomials

36. Factoring When a∙c is Positive

37. Factoring When a∙c is Negative

38. Factoring Special Cases
8.7
Factoring Special Cases

39. Essential Understanding
You can factor some trinomials by “reversing” the rules for multiplying special case binomials that you learned in lesson 8.4

40. Factoring Perfect Square Trinomials
Any trinomial of the form a² + 2ab + b² is a perfect square trinomial because it is the result of squaring a binomial.
For example let a,b be real numbers:
(a + b)² = (a + b)(a + b) = a² + 2ab + b²
(a - b)² = (a - b)(a - b) = a² - 2ab + b²

41. Factoring a Difference of Two Squares
For all real numbers a,b:
a² - b² = (a + b)(a – b)

42. 8.8
Factoring by grouping

43. Essential Understanding
Some polynomials of a degree greater than 2 can be factored

44. Factoring Polynomials
Factor out the greatest common factor (GCF)
If the polynomial has two terms or three terms, look for a difference of two squares, a perfect square trinomial or a pair of binomial factors
If the polynomial has four or more terms, group terms and factor to find common binomial factors.
As a final check, make sure there are no common factors other than 1