Name: ________________________

Algebra

Summer Review Packet

About Algebra 1: Algebra 1 teaches students to think, reason, and

communicate mathematically. Students use variables to determine

solutions to real world problems. Skills gained in Algebra 1 provide

students with a foundation for subsequent math courses. Students

use a graphing calculator as an integral tool in analyzing data and

modeling functions to represent real world applications. Each

student is expected to use calculators in class, on homework, and

during tests.

Expectations for the Summer Packet: The problems in this packet are

designed to help you review topics that are important to your

success. All work must be shown for each problem. The problems

should be done correctly, not just attempted. The packet is due the

first day of school. During the first week of school, concepts in the

packet will be reviewed.

Table of Contents

Radicals

Exponents

Solving Linear Equations and Inequalities

Graphing Linear Equations

Adding, Subtracting, and multiplying polynomials

Factoring

Solving Systems

Radicals

Simplify the following radicals. (Hint… Write each number as factors,

then look for pairs.)

√121

√12

√16

√100

(√144)2

√336

√210

(√9)3

√8

√19

Exponents

Simplify each expression. Assume all variables are nonzero.

(−3𝑎2𝑏3)2 c3d2(c-2d4)

5𝑢𝑣6

𝑢2𝑣2

10( 𝑦5

𝑥2 )2

-2s-3t(7s-8t5)

-4m(mn2)3

(4𝑏)2

2𝑏

𝑥−1𝑦−2

𝑥3𝑦−5

Solving Linear Equations and Inequalities

Examples…

Solve for x. Show all supporting work.

-2(x + 3) = 4x - 3 7x – 17 = 4x + 1

5 – x – 2 = 3 + 4x + 5 5 – (x – 4) = 3(x + 2)

2(x + 4) – 5 = 2x + 3 3(2x – 1) + 5 = 6(x + 1)

Examples…

Solve for x… Show supporting work.

4x – 9 > 7 -5 > -5 – 3x

-4(x + 3) > 24 4 > x – 3(x + 2)

Graphing Linear Equations

Steps for Graphing Linear Equations

1. Put the equation in slope intercept form: y=mx+b

2. Graph the b value on the y-axis

3. Use the slope, m, to move the point you made up or down and

right or left

If m is negative move the numerator value down, if it is positive

move it up

Use the denominator to move the point right

4. Connect the dots

1. y= 3x+4

2. y= 2

3𝑥+5

3. y-3= 5x

4. y-5= 3

4𝑥 + 2

Adding, Subtracting, and Multiplying polynomials

Ex: Add the following polynomials. Remember to group like terms.

Ex: Subtract the following polynomials. Remember to group like

terms.

Ex: Multiply the following polynomials. Remember to group like terms.

Simplify the following polynomials.

(𝑥2 + 𝑥 + 7) + (3𝑥2 + 2𝑥 + 1)

(𝑥2 + 5𝑥 − 3) + (11𝑥2 − 2𝑥 + 8)

(8𝑥2 + 3𝑥 − 3) − (2𝑥2 − 5𝑥 + 9)

(7𝑥2 − 2𝑥 − 3) − (2𝑥2 − 6𝑥 + 5)

x(3𝑥2 + 2𝑥 − 11)

3x(6𝑥2 − 4𝑥 + 5)

Factoring

A composite number is a number that can be written as the product of two

positive integers other than 1 and the number itself. For example: 14 is a

composite number because it can be written as 7 times 2. In this case, 7 and 2

are called factors of 14.

A composite expression is similar in that it can be written as the product of two

or more expressions. For example: x2 + 3x + 2 is composite because it can be

written as (x + 1)(x + 2). (Recall that the FOIL Method shows that (x + 1)(x + 2) is

equivalent to x2 + 3x + 2.) Here, (x + 1) and (x + 2) are factors of x2 + 3x + 2.

In general, a number is a factor of another number if the first number can divide

the second without a remainder. Similarly, an expression is a factor of another

expression if the first can divide the second without a remainder.

Definition

A prime number is a number greater than 1 which has only two positive factors:

1 and itself. For example, 11 is a prime number because its only positive factors

are 1 and 11.

Factoring is a process by which a the factors of a composite number or a

composite expression are determined, and the number or expression is written

as a product of these factors. For example, the number 15 can be factored into:

1 * 15, 3 * 5, -1 * -15, or -3 * -5. The numbers -15, -5, -3, -1, 1, 3, 5, and 15 are all

factors of 15 because they divide 15 without a remainder.

Factoring is an important process in algebra which is used to simplify expressions,

simplify fractions, and solve equations. The next few lessons explain how to

factor

Ex: Find the greatest common factor of the given functions.

Ex: Factor the given functions.

Find the greatest common factor of the following functions:

2x2 + 4x + 6

4x2 – 12x + 4

3x2 – 9x – 12

15x2 +30x - 45

x3 + 4x2 + 17x

-2x -14x3

Factor the following functions:

5x2 + 17 + 6

4x2 + 16x + 15

4x2 – 33x + 8

-6x2 + 11x - 4

6x2 – 7x – 20

x2 + 6x + 9

Solving Systems

Ex: Solve the system using substitution

Ex: Solve the system using elimination

Solve the following problems using the substitution method.

2x + 3y = 17

x + y = 11

5x – 7y = 12

x – y = 33

Solve the following problems using the elimination method.

2x – 2y = 13

3x + y = 6

3x + 2y = 8

3x + 5y = 12

Dividing Polynomials

Use long division.

Use synthetic division.

Finding all Zeros of a Function

Find all of the zeros/roots.

Operations with Complex Numbers

Add, subtract or multiply.

Logarithmic Functions

Express as a single logarithm.

Simplify, if possible.

Solve.

Rational Expressions

Simplify, add, subtract, multiply, and/or divide.

Solve.