Algebra III
Summer Assignment 2015
The following packet contains topics and definitions that you will be required to know in order to
succeed in Algebra III this year. You are advised to be familiar with each of the concepts and to
complete the included problems by Thursday, September 3, 2015. All of these topics were discussed
in either Algebra II or Geometry and will be used frequently throughout the year. All problems
that you are to complete are marked in bold. All problems are expected to be completed.
Section I: Lines
Lines:
𝑦 −𝑦
Slope:
𝑚 = 𝑥2 −𝑥1
Slope Intercept Form:
y = mx + b
Standard Form:
ax + by = c
Point-Slope Form:
y - y1 = m(x – x1)
2
1
Intercepts:
In order to find the x-intercept(s) of an equation, you have to set y equal to zero and solve the
equation for x. In order to find the y-intercepts of an equation, you have to set x equal to zero
and solve the equation for y.
Parallel Lines:
Two lines whose graphs have the same slope.
Perpendicular Lines:
Two lines whose graphs have opposite reciprocal slopes.
Find the slope of the lines passing through each set of points:
𝟏
(𝟔, 𝟏𝟐) 𝒂𝒏𝒅 (−𝟔, −𝟐)
Find the slope of the line:
𝟏
𝟏
(− 𝟑 , 𝟎) 𝒂𝒏𝒅 (− 𝟐 , − 𝟐)
Find the F
Find the slope of the following Line:
y
𝟑𝒙 + 𝟓𝒚 = −𝟏𝟓
4
2
–4
–2
O
–2
–4
2
4
x
Write the equation of the line in standard
Find the point-slope form of the line
form for a slope of -8 and through (-2, -2)
through (-6, -4) and (2, -5).
Write the equation of the line in slope-intercept
Write the equation of the horizontal
form through (−𝟏𝟎, −𝟑) and (−𝟏, 𝟏).
line through (−𝟑, −𝟏).
Graph the equation of 6x + 5y = 30
using the x- and y-intercepts.
𝟑
𝟒
Graph 𝒚 = − 𝒙 − 𝟐
Find the equation of the line that is parallel
Find the equation of the line perpendicular
to 𝒚 = − 𝟑𝒙 + 𝟒 and goes through (−𝟒, 𝟔).
to 𝒚 = − 𝟒 𝒙 + 𝟏 and goes through (2, 6).
𝟓
Give the slope-intercept form for the equation of the line that is perpendicular to 𝟕𝒙 + 𝟑𝒚 = 𝟏𝟖 and
contains (𝟔, 𝟖).
Which two lines are parallel?
I.
II.
III.
What must be true about the slopes of two perpendicular lines, neither of which is vertical?
Section II: Polynomials
Properties of Exponents:
𝑥 𝑛 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ … ∙ 𝑥 (n factors of x)
1. Whole number exponents:
𝑥 0 = 1, 𝑥 ≠ 0
2. Zero exponents:
𝑥 −𝑛 =
3. Negative Exponents:
1
𝑥𝑛
𝑛
√𝑥 = 𝑎 → 𝑥 = 𝑎 𝑛
4. Radicals (principal nth root):
5. Rational exponents:
𝑥
1⁄
𝑛
6. Rational exponents:
𝑥
𝑚⁄
𝑛
𝑛
= √𝑥
𝑛
= √𝑥 𝑚
Operations with Exponents:
𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑚+𝑛
1. Multiplying like bases:
𝑥𝑚
2. Dividing like bases:
𝑥𝑛
= 𝑥 𝑚−𝑛
𝑥 𝑛
𝑥𝑛
3. Removing parentheses: (𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛 ( ) = 𝑛 (𝑥 𝑛 )𝑚 = 𝑥 𝑛𝑚
𝑦
𝑦
For example:
Simplify each of the following expressions:
a.
7a  2a 
2
5
 7 2 a ( 2 5)
 14a 3
 14
 3
a
b.
 2 x
1
y2

  y 
  2  x 1
3
3
 8 x y

 8y
x3
6
c.
3
6
3
2ab 5 c 2
a 3bc 2
2 3
 2a (13)b ( 51) c ( 2  2 )
 2a  2b 4 c 0

2b 4
a2
Simplify the following expressions. Use only positive exponents.
1.
3a 4a 
2.
 4x  2x 
2
6
13.
x 4 x 2
x 5
2
2
14.
12 x y 
15.
4 p q p q 
16.
4x3
2x
17.
p 
18.
 15 x 4
3x
19.
r 2 s 3t 4
r 2 s 4 t 4
20.
xy 2 6 x

2 y2
21.
s t  st 
2
3.
4 x y 
4.
2 x
5 2
3
5
y

4 3
5
5.
6.
8a
2a 2
6x 7 y 5
3 x 1
4 x 
2 0
7.
8.
9.
2 xy5
 3x 2 


 2 
2
 6m n 3mn
2
6 2
8x 4 y 7
2
2
2 2
2
10. 3x 4 y 5 
2
3
3
22. 3x 3 y 2 
2
11.
2r

1 2 0 2
s t
2rs
23.
12. x 5 2x 3
24.
h k 
4
5 0
s 2t 3 sr 3

r
t
3
Section III: Radical Expressions and Complex Numbers
Radical Expressions:
𝑛
𝑛
𝑛
𝑛
𝑛
Multiplying Radical Expressions: If √𝑎 𝑎𝑛𝑑 √𝑏 are real numbers, then √𝑎 ∙ √𝑏 = √𝑎𝑏
𝑛
𝑛
𝑛
Dividing Radical Expressions: If √𝑎 𝑎𝑛𝑑 √𝑏 are real numbers and b≠ 0, then
√𝑎
𝑛
𝑎
= √𝑏
√𝑏
𝑛
For example:
Simplify each of the following rational expressions (be sure to rationalize the denominator if
necessary):
√𝑥 3
√5𝑥𝑦
First use the Dividing Radical Expressions rule to
rewrite with one radical.
√
Next cancel any common factors in the numerator and
denominator.
𝑥3
5𝑥𝑦
𝑥2
√
5𝑦
√𝑥 2
Now use the Dividing Radical Expressions rule to
rewrite with two radicals.
√5𝑦
𝑥
Simplify any possible square roots.
√5𝑦
𝑥 √5𝑦
∙
√5𝑦 √5𝑦
Rationalize the denominator to get you final answer.
𝑥√5𝑦
5𝑦
Final Answer
Simplify the following expressions. Use only positive exponents.
1)
√2
5)
√3
2
2) √3𝑥
6)
√2𝑥 3
7)
3)
4)
√10𝑥𝑦
15√60𝑥 5
3√12𝑥
8)
√3𝑥𝑦 2
√5𝑥𝑦 3
√5𝑥 4 𝑦
√2𝑥 2 𝑦 3
10
√5𝑥 2
3√11𝑥 3 𝑦
−2√12𝑥 4 𝑦
Imaginary Numbers:
For any positive real number a, √𝑎 = 𝑖 √𝑎
𝑖 2 = −1
𝑜𝑟 √−1 = 𝑖
For example:
Simplify √−8 by using the imaginary number i.
√−8
First factor out the -1 from -8
√−1 ∙ 8
𝑖√8
Next replace √−1 with i
𝑖√4 ∙ 2
Then factor any perfect squares.
2𝑖√2
Take the square root of the perfect square.
Simplify the following by using the imaginary number i.
1) √−12
4) √−68
2) √−20
5) √−1
3) √−300
6) √−121
Complex Numbers:
A complex number can be written in the form 𝑎 + 𝑏𝑖, where a & b are real numbers, including 0.
For example:
Simplify the expression (5 + 7𝑖) + (−2 + 6𝑖)
(5 + 7𝑖) + (−2 + 6𝑖)
Use the Commutative and Associative Properties of Addition to rewrite
expression with like terms together
(5 + −2) + (7𝑖 + 6𝑖)
(3 + 13𝑖)
Simplify
For example:
Simplify the expression (5𝑖 + 1)(−4𝑖)
(5𝑖 + 1)(−4𝑖)
(5𝑖)(−4𝑖) + 1(−4𝑖)
Use the Distributive Property to rewrite expression.
−20𝑖 2 − 4𝑖
Multiply
Replace 𝑖 2 with -1
−20(−1) − 4𝑖
20 − 4𝑖
Simplify
Simplify the following by using the imaginary number i.
7) (2 + 4𝑖) + (4 − 𝑖)
10) 6 − (8 + 3𝑖)
8) (12 + 5𝑖) − (2 − 𝑖)
11) (8 + 𝑖)(2 + 7𝑖)
9) (−2𝑖)(5𝑖)
12) (9 + 4𝑖)2
Section IV: Quadratic Equations
Standard Form of a Quadratic Equation: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Sum and Difference of Same Terms: (𝑢 + 𝑣)(𝑢 − 𝑣) = 𝑢2 − 𝑣 2
Square of a Binomial: (𝑢 + 𝑣)2 = 𝑢2 + 2𝑢𝑣 + 𝑣 2 and (𝑢 − 𝑣)2 = 𝑢2 − 2𝑢𝑣 + 𝑣 2
For example:
Write the following quadratic equations in standard form, then identify a, b, and c.
2
First move the −3𝑥 to the other side of the equation by
adding 3𝑥 2 to both sides of the equation
Now move the 2 to the other side of the equation by
subtracting 2 from both sides.
So a=3 b=4 and c=-2
4𝑥 = 2 − 3𝑥 2
+3𝑥 2
+ 3𝑥 2
3𝑥 2 + 4𝑥 = 2
-2
-2
3𝑥 2 + 4𝑥 − 2 = 0
Write the following quadratic equations in standard form, then identify a, b, and c.
1) 2𝑥 2 − 11𝑥 = −15
4) 2𝑥 2 − 1𝑥 = −5 − 3𝑥
2) 𝑥 2 + 7𝑥 = 18
5) −7𝑥 = 6𝑥 2 − 4
3) 16𝑥 2 = 8𝑥
6) −14𝑥 − 2𝑥 2 = 6𝑥 2 + 3𝑥
For example:
Solve the following quadratic equations by factoring:
𝑥 2 + 5𝑥 + 4 = 0
To factor this quadratic equation, you first must
split the 𝑥 2 term.
(𝒙
) (𝒙
)
=𝟎
Now you must find two numbers that multiply to
equal c (4) Either 1 * 4 or 2 * 2 will work.
You must choose the set of numbers that and add to
equal b (5) In this case 1 & 4
Set each set of parenthesis = 0 to and solve
(𝒙 + 𝟏 ) (𝒙 + 𝟒)
=𝟎
(𝒙 + 𝟏 ) = 𝟎 𝒐𝒓 (𝒙 + 𝟒 ) = 𝟎
𝒙 = −𝟏
𝒐𝒓
𝒙 = −𝟒
Solve the following quadratic equations by factoring:
Since the c=0 in the equation, you factor this quadratic
by taking out the GCF
−3𝑥 2 + 12𝑥 = 0
The GCF of both terms is -3x, so factor this out of
BOTH terms. (Note: you can check you factored
correctly by re-distributing -3x, you should get the
original equation.)
(−𝟑𝒙)(𝒙 − 𝟒) = 𝟎
Lastly, set each set of parenthesis = 0 to and solve
−𝟑𝒙 = 𝟎 𝒐𝒓 𝒙 − 𝟒 = 𝟎
𝒙 = 𝟎 𝒐𝒓 𝒙 = 𝟒
Solve each of the following by Factoring:
7) 𝟑𝒙𝟐 − 𝟔𝒙 = 𝟎
11) 𝒙𝟐 − 𝟏𝟑𝒙 + 𝟒𝟐 = 0
8) 𝟒𝒙𝟐 + 𝟐𝟎𝒙 − 𝟏𝟐 = 𝟎
12) 𝒙𝟐 − 𝟐𝒙 − 𝟑𝟓 = 𝟎
9) 𝟗𝒙𝟐 − 𝟒𝟐𝒙 + 𝟒𝟗 = 𝟎
13) 𝟒𝒙𝟐 − 𝟏𝟒𝟒 = 0
10) 𝟑𝒙𝟐 − 𝟏𝟔𝒙 + 𝟓 = 𝟎
𝟏𝟒) 𝒙𝟐 + 𝟗 = 𝟎
Solve each of the following by using the Quadratic Formula: 𝑥 =
15) 𝒙𝟐 + 𝟐𝒙 + 𝟒 = 𝟎
17)
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
16)
𝟑𝒙𝟐 + 𝟐𝒙 + 𝟏 = 𝟎
𝟖𝒙𝟐 − 𝟏𝟎𝒙 − 𝟑 = 0
18) 𝒙𝟐 − 𝟐𝒙 − 𝟑 = 𝟎
Expand each of the following (hint: FOIL):
19) (𝒙 + 𝟒)𝟐
21)
(𝒙 − 𝟐)𝟑
19) (𝒙 − 𝟒)(𝒙 + 𝟑)
22) (𝒙 − 𝟔)(𝒙 − 𝟑)