SAGE review 2
(Units 4 – 6)
Notes
The Remainder Theorem
When any polynomial or function, 𝑓(𝑥), is divided by a
linear factor (𝑥 − 𝑟), the remainder is 𝑅 = 𝑓(𝑟), or the
value of the equation or function when 𝑥 = 𝑟
The Factor Theorem
Forms of Lines
Point-Slope Form: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Slope-Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏
Rational Root Theorem
factors of the constant
A polynomial has a linear polynomial as a factor if and
only if the remainder is zero, or in other words, 𝑓(𝑥) has
(𝑥 − 𝑟) as a factor if and only if 𝑓(𝑟) = 0
Difference of Two Squares:
𝑎2 − 𝑏 2 = (𝑎 − 𝑏)(𝑎 + 𝑏)
Possible Rational Roots: ± factors of the leading coefficient
Scatter Plot on a Calculator
Enter your data into two lists: STAT → ENTER → enter
the data into two lists, then 2nd → Y= (STAT PLOT) →
ENTER → ON → scatter plot → enter the specific lists
where your data is listed
Sum or Difference of Two Cubes
Set an appropriate window → GRAPH
𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 )
𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )
Quadratic Regression on a Calculator
STAT then ENTER then enter your points into
Quadratic Function Forms
Vertex Form: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
vertex: (ℎ, 𝑘)
Factored Form: 𝑓(𝑥) = 𝑎(𝑥 − 𝑟1 )(𝑥 − 𝑟2 )
zeroes/roots: (𝑟1 , 0), (𝑟2 , 0)
Standard Form: 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Imaginary Numbers
√−1 = 𝑖
𝑖 2 = −1
Complex Conjugates
Pairs of numbers of the form, 𝑎 + 𝑏𝑖 and 𝑎 − 𝑏𝑖
Quadratic Formula
𝑥=
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
Discriminant
𝑏 2 − 4𝑎𝑐
two lists (remember where)
Then: STAT → CALC
then choose your lists/function location (separated by a
coma in the old operating system)
Turning on 𝑹𝟐
MODE then “Stat Diagnostics” ON
Or
Catalog and scroll down to “Diagnostics On” ENTER
When simplifying rational expressions or +, −,×,÷
rational expressions:
ALWAYS FACTOR FIRST! NEVER CANCEL
TERMS, ONLY CANCEL FACTORS!
You don’t need a common denominator to multiply or
divide rational expressions, you DO need a common
denominator to add or subtract rational expressions.
Radical Equations
Inverse Functions
To solve radical equations, you need to “undo” the
radical. To undo a square root, you square; to undo a
cube root, you cube; etc. You must isolate a single
radical on one side of the equation first.
The inverse function of f , which is symbolized 𝑓 −1, is
the function containing all of the ordered pairs of f with
the first and second coordinates switched. For example, if
(𝑎, 𝑏) is contained in f , then (𝑏, 𝑎) is contained in 𝑓 −1.
This means that the domain of f equals the range of 𝑓 −1,
and the range of f equals the domain of 𝑓 −1
You must also be aware that raising both sides of an
equation to an even power (to “undo” radicals with
even root indices) can create extraneous solutions.
Thus, you must check your solutions to such
equations.
* To be a function it must pass the Vertical Line Test.
To be an invertible function (it has an inverse) it must also
When equations involve expressions having fractional pass the Horizontal Line Test. If it passes BOTH the
exponents, you should realize that these equations are Vertical and Horizontal Line Tests then it’s a One-to-One
merely radical equations “in disguise.” If you convert Function.
the equations to radical equation from, you can use
the techniques for solving radical equations.
Finding an inverse
1. Write the function in the form y = f(x)
Properties of Exponents
1.
𝑥 𝑚 ⋅ 𝑥 𝑛 = 𝑥 𝑚+𝑛
𝑥𝑚
2.
𝑥𝑛
= 𝑥 𝑚−𝑛
2. Switch x and y.
3.
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
4.
𝑥 −𝑛 =
1
𝑥𝑛
1
and
𝑥 −𝑛
= 𝑥𝑛
3. Solve for y.
5.
𝑥 0 = 1, 𝑥 ≠ 0
(00 is undefined.)
Properties of Radicals
1.
3.
1
𝑛
√𝑥 = 𝑥 𝑛
𝑛
𝑥
√𝑦 =
𝑛
√𝑥
√𝑦
𝑛
𝑛
2. 𝑛√𝑥𝑦 = √𝑥 ⋅ 𝑛√𝑦
𝑛
𝑚
𝑛
4. √𝑥 𝑚 = 𝑥 𝑛 = ( √𝑥)
4. Replace y with 𝑓 −1 (𝑥)
𝑚
SECONDARY III
SAGE review 2 (units 4-6)
Name: _____________________________
Period: _________________
Completely factor each of the following expressions.
1. x2 – 5x – 24
2. 3x2 + 20x – 7
4. 6x3 – 9x2 – 15x
5. x3 + 3x2 – 4x – 12
Write the zero that corresponds to each factor.
6. x – 8
7. 3x + 7
3. 125x3 – 64
Write the factor that corresponds to each zero.
4
8. x = 5
9. 𝑥 = − 3
Use long division to determine the quotient (and remainder, if applicable).
10. (x3 + 12x2 + 17x – 30) ÷ (x – 3)
Use synthetic division to determine the quotient (and remainder, if applicable). Don’t forget to write the
polynomial for the quotient once you have done the synthetic division.
11. (x4 − 4x3 + 8x2 – 16x + 16) ÷ (x – 2)
Determine all possible rational roots of each polynomial using the Rational Root Theorem.
12. x4 – 4x3 + 9x2 – 20x + 20 = 0
13. Brian wanted to determine the relation that might exist between speed and miles per gallon of an automobile. Let x be
the average speed of a car on the highway measured in miles per hour and let y represent the miles per gallon of the
automobile. The following data are collected:
x
50
55
55 60 60 62 65 65
y
28
26
25 22 20 20 17 15
A. Write a linear regression model for the miles per gallon in terms of speed.
B. Use your model to predict the miles per gallon of a car traveling 61 miles per hour.
C. How fast is a car traveling if it is getting 30 miles per gallon?
D. Explain what the slope represents in the context of the data.
Solve each inequality. Sketch the solution on a number line then write your solution in inequality form and
interval notation.
14. 2(x – 4) ≥ 4x + 6
16.(x – 1)2(x + 3)(x + 1) ≥ 0
15. x2 + 7x < -12
Simplify each problem. Express answers in simplest form (leave in factored form; do NOT multiply out). List
ALL DOMAIN RESTRICTIONS. Show work.
17.
15𝑚𝑛7
10𝑚4 𝑛4
18.
𝑥 2 +5𝑥+6
𝑥 2 −4
20.
𝑥 2 +2𝑥−8
𝑥−2
÷ 3𝑥+3
𝑥 2 +4𝑥+3
21.
𝑥
𝑥+1
+ 3𝑥+6
𝑥 2 −4
19.
22.
𝑥 2 +2𝑥−3 𝑥 3 +𝑥 2
∙
𝑥2
𝑥+3
𝑥
𝑥 2 +5𝑥+6
2
− 𝑥 2 +4𝑥+4
Solve each equation. Show all work. Simplify all answers. Remember to check for extraneous solutions.
23.
2x2 = 19x + 33
24.
2
𝑥 2 −6𝑥+8
1
2
= 𝑥−4 + 𝑥−2
Graph the piecewise function and evaluate the function at each specified
value.
25.
 1
 x  4,
f ( x)   2
( x  2) 2 ,

a. f (2)
b. f (0)
x0
x0
c. f (-2)
d. f (1)
SIMPLIFYING EXPRESSIONS: Simplify the following expressions as far as possible by performing the indicated
operation(s). Do not leave negative exponents or exponents that are 0 (zero).
26. 𝑥 4 𝑥 3
27. (−3𝑥𝑦 5 )(4)
28. (2𝑥 3 )(4𝑥)2
29. (𝑦 5 )3
30. (4𝑚𝑛3 )2
31. (
32. (4𝑥 − 3)2
33.
4
(2𝑦 −3 𝑧)
(4𝑦𝑧 −1 )3
−4
2𝑥 3
)
𝑦
34. (5𝑐 −4 𝑑0 )3
SIMPLIFYING RADICALS: Simplify each radical. Combine like terms (if needed and if possible). Remember:
no radicals can remain in a denominator.
25
36. √81
35. √98
1
37. 3√75
1
−3
38. √−36𝑥
39. 812
40. (−125𝑥 6 𝑦 5 )3
41. 16 4
43. 5√27 − √48
44. 2√3(4√3 + √2)
45. (7 + √5)(3 − 4√2)
2
46. (2 − 3√5)
47.
4+√2
√5
48.
3+√2
√2−√5
42. 5√2 + 4√3 − 12√2
SOLVE EACH RADICAL EQUATION. Remember to check your answers AND you must show your work in
checking the answers.
49.
4 2 x  6 1  15
51. √7𝑥 − 7 = √3𝑥 − 2
53.
50.
3
6u  5  2  3
52. 𝑥 − 3 = √𝑥 + 3
A tennis ball is hit upward with an initial velocity of 42 feet per second. This can be modeled by the function
h(t )  16t 2  42t
where h(t) is the height and t is the time in seconds. How long does it take for the ball to hit the
ground?
54. Given the function f ( x)  ( x  2)( x  1) 2 ( x  5) 3 Find:
a. The degree of the polynomial
b. End behavior
c. Zeros of the polynomial
d. Graph the function
55. Sketch the graph of f ( x)  x3  9 x by finding:
a. End behavior:
b. y-intercept
c. x-intercepts