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. ( 3a 2 b 3 ) 2 c 3 d 2 (c -2 d 4 ) 5uv 6 u 2 v 2 y5 10( x 2 )2-2s -3 t(7s -8 t 5 ) -4m(mn 2 ) 3 (4b) 2 2b x 1 y 2 x 3 y 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 3 x+5
3. y-3= 5x 4. y-5= 3 4 x + 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. (x 2 + x + 7) + (3x 2 + 2x + 1) (x 2 + 5x 3) + (11x 2 2x + 8) (8x 2 + 3x 3) (2x 2 5x + 9) (7x 2 2x 3) (2x 2 6x + 5) x(3x 2 + 2x 11) 3x(6x 2 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: x 2 + 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 x 2 + 3x + 2.) Here, (x + 1) and (x + 2) are factors of x 2 + 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: 2x 2 + 4x + 6 4x 2 12x + 4 3x 2 9x 12 15x 2 +30x - 45 x 3 + 4x 2 + 17x -2x -14x 3 Factor the following functions: 5x 2 + 17 + 6 4x 2 + 16x + 15 4x 2 33x + 8-6x 2 + 11x - 4 6x 2 7x 20 x 2 + 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.