1. Chapter 5/6/7
Polynomials

2. 6.1 Operations with Polynomials
Monomial- number, variable, or product of a number and one or more variables
Cannot have: variables in denominator, variables with negative exponents, or variables under radicals
Constants-numbers with no variables
Coefficient – Numerical factor of monomial
Degree- sum of the exponents of its variables
Examples

3. Negative Exponents

4. Product of Powers
Add exponents
Examples:

5. Quotient of Powers
Subtract exponents
Examples:

6. Power of a Power
Multiply exponents
Examples

7. Power of a Product
Raise each factor to the exponent
Examples

8. Power of a Quotient
Raise top and bottom by exponent
Examples

9. Simplifying using several properties
Examples

10. Polynomials Polynomial- Monomial or sum of monomials
Each monomial is a term of the polynomial
Binomial- two terms
Trinomial- three terms
Degree of Polynomial- degree of term with highest degree
3x2 + 7
3x5y3 – 9x4
Like terms – x and 2x

11. Simplifying Polynomials
Distribute across grouping symbols and combine like terms
Examples

12. Multiplying Binomials
FOIL
First, Outer, Inner, Last
Box
Examples

13. Multiplying Polynomials
Distributive property and combine like terms
Examples

14. 6.2 Dividing Polynomials

15. Polynomial by a Monomial
Break into sum of separate fractions and simplify
Examples

16. Polynomial by Binomial
Long Division
Synthetic Division
Examples

17. 5.3 Factoring Polynomials
Always look for GCF first!

18. Factoring binomials (2 terms)
Difference of two squares
a² - b² = (a + b)(a – b)
Difference of two cubes
a³ - b³ = (a – b)(a² + ab + b²)
Sum of two cubes
a³ + b³ = (a + b)(a² - ab + b²)

19. Factoring trinomials (3 terms)
Perfect Square Trinomials
a² +2ab+b² = (a + b)²
a² -2ab+b² = (a - b)²
Factoring Trinomials
Grouping  Four terms
Ax+bx+ay+by=x(a+b)+y(a+b)(x+y)(a+b)

20. Simplifying Quotients
Factor numerator and denominator completely
Simplify by cancelling

21. 7.4 Roots of Real Numbers
Square root
Cube root
nth root

22. Index-Radical-Radicand

23. Principal Root
When there is more than one real root, the non-negative root is the principal root
Examples

24. Real Roots n b>0 b<0 b=0 even One + One - No real roots
One real root, 0
odd
No -
One –
No +

25. Simplifying Square Roots
Rules for Simplifying Radicals
There are no perfect square, cubes, etc. factors other than 1 under the radical.
There isn’t a fraction under the radical.
The denominator does not contain a radical expression.

26. 7.5 Radical Expressions Product Property Quotient Property
Rationalize the denominator means get the radical out of the bottom
Like radical expressions have the same index and same radicand

27. Radicals simplified if
Index small as possible
No more factors of n in radicand
No more fractions in radicand
No radicals in denominator

28. 7.6 Rational Exponents
Two big rules

29. Examples
X1/5*x7/5
X-3/4
X-2/3
6√16
3√2
6√4x4
Y1/2+1
Y1/2-1

30. 7.7 Radical equations and inequalities
Variables in radicand
Undo square roots by squaring
Undo cube roots by cubing
Check for extraneous solutions
A number that does not satisfy the original equation
ALWAYS go back and check solutions
Inequalities: must also solve for
Radicand ≥ 0

31. Examples √x+1)+2=4 √x-15)=3- √x 3(5n-1)1/3-2=0 2+ √4x-4)≤6 √3x-6)+4≤7
Solution does not check  No real solutions
3(5n-1)1/3-2=0
2+ √4x-4)≤6
Solve 4x-4≥0 first because even index so can’t be negative
Then solve complete inequality and test intervals
√3x-6)+4≤7

32. On calculator Set everything = 0 Graph Find “zero” (2nd calc 2:zero)
If inequality look for interval that is above or below the x-axis

33. 5.4 Complex Numbers
Big rules for Imaginary unit i

34. Powers of i √-18 √-32y3 -3i*2i √-12* √-2 Divide exponent by 4
Use remainder as new exponent
Simplify i35
Simplify i55

35. Complex number
a + bi
a is real part, b is imaginary part

36. To find values of x and y that make equations true…
Set real parts equal and solve
Set imaginary parts equal and solve
2x-3+(y-4)i=3+2i

37. To add/subtract Add real parts Add imaginary parts (7-6i)+(9+11i)

38. To multiply complex numbers
FOIL/Box and then simplify
(6-4i)(6+4i)
(4+3i)(2-5i)
(7+2i)(9-6i)

39. Dividing complex numbers
Multiply top and bottom by conjugate of bottom
2/(7-8i)
(3-i)/(2-i)
(2-4i)/(1+3i)

40. Solving equations using i
5x2+35=0
-5m2-65=0
-2m2-6=0
Find values of m and n to make equation true
(3m+4)+(3-n)i=16-3i