Accuplacer Review Workshop Elementary Algebra Part II Week Three Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com http://www.coolmath.com http://www.testprepreview.com http://www.kahnacademy.org Preparing for the Algebra Test: To Place into College Level Math It is important to know that the accuplacer assesses three areas of Algebra: Numerical Skills/Pre-Algebra Elementary Algebra Intermediate Algebra These areas are required prerequisites for College Algebra. The following workshop will hopefully aide you in your review of Elementary Algebra. NOTE: Although some questions are basic skills many accuplacer questions require you to put several math concepts together, or work multiple step problems to determine the final answer. 1
Elementary Algebra Part II: Questions in the Elementary Algebra range from introductory algebra concepts and skills to the knowledge and skills considered necessary to enter an intermediate algebra course. The required skills also cover a majority of items from these following categories: Simplifying Radicals (Using multiplication and division properties) Basic Operations of Radicals 1. Addition 2. Subtraction 3. Multiplication Basic Operation of Polynomials 1. Addition 2. Subtraction 3. Multiplication 4. Division 2
Simplifying Radicals: To simplify a radical means to make the number inside the radical as small as possible (but still a whole number). This is done by using the multiplication or division properties of radicals. Multiplication Property A Product of Two Radicals with the same Index Number x n y = xy In other words, when you are multiplying two radicals they must have the same index number. Then you can write the product under one radical with the common index. Note: This rule can be used in either direction Example 1: Use the product rule to simplify 8 8 = (4)(2) = 4 2 = 2 2 Example 2: Use the product rule to multiply 3 2 3 2 = (3)(2) = 6 Since we cannot take the cube root of 6 and 6 does not have any factors we can take the cube root of, this is the simplified answer. Example 3: Use the product rule to multiply 2x²y 5xy 2x²y 5xy = (2x y)(5xy) = 10x³y² Since we cannot take the fourth root of any of the variables inside the radical sign and 10 does not have any factors with a fourth root this is the simplified answer. 3
Quotient Property A quotient of Two Radicals with the same Index Number If n is even, x and y represent any nonnegative real number and y does not equal zero. If n is odd, x and y represent any real number and y does not equal zero. = Note: This rule can also work in either direction. Example 4: Use the quotient rule to simplify " = = " " Since we cannot take the square root of 2 and 2 does not have any factors that we can take the square root of, this is the simplified answer. Example 5: Use the quotient rule to simply = = Since we cannot take the cube root of 5 and 5 does not have any factors that we can take the cube root of, this is the simplified answer. Example 6: Use the quotient rule to divide then simplify " " = " = 10 Since we cannot take the square root of 10 and 10 does not have any factors we can take the square root of, this is the simplified answer. 4
Basic Operations of Radicals: Like Radicals Like radicals are radicals that have the same root number AND radicand (expression under the radical sign). Example: xy and 5xy -2 4ab and 7 4ab Adding and Subtracting Radical Expressions Step 1: Simplify the radicals Step 2: Combine like radicals Example 7: Add 2 20x + 3 5x First Simplify: 2 (4)(5x) + 3 5x 2 4 5x + 3 5x 2 (2) 5x + 3 5x 4 5x + 3 5x Collect like radicals 7 5x Example 8: Combine like radicals 3b³ - 3b 24 + 2 81b³ First Simplify: b 3 Collect like radicals b 3 6b 3 + 6b 3 5
Example 9: Add the radicals First Simplify: " " + + + Combine like radicals = x Step 1: Multiply the radical expression Step 2: Simplify the radicals Multiply Radical Expressions Example 10: Multiply and Simplify 2 ( 3-8) Step 11: Multiply the radical expression AND Step 2: Simplify 2 ( 3-8) = 6-16 = 6 4 Example 11: Multiply and Simplify ( a 5) (3 a + 7) Step 12: Multiply the radical expression AND Step 2: Simplify ( a - 5) (3 a + 7) = 3 a² + 7 a - 15 a 35 = 3a - 8 a 35 6
Polynomials: A polynomial is a finite sum of terms separated by + and - signs and has constants, variables and exponents 0,1,2,3. but it never has division by a variable. Example: 3x - 5x + x - 10 Degree of the Polynomial The degree of the polynomial is the largest degree of all its terms Example: -4x + 7x -3 The polynomial has a degree of 4 Monomial, Binomial, Trinomial These are special names for polynomials with 1, 2, or 3 terms: Example: 3xy² is a monomial (1 term) 5x 1 is a binomial (2 terms) 3x + 5y² = 3 is a trinomial (3 terms) Combining like Terms Like Terms are terms that have the exact same variables raised to the exact same exponents Example: 3x - 5x Simplify by combining like terms: 5x + 7x - 2x -10x + 5 = 3x 3x + 5 7
Basic Operations of Polynomial: Step 1: Remove the parenthesis Adding Polynomials If there is only a + sign in front of ( ), this is multiplication by positive one so all the terms inside of the ( ) remain the same when you remove the parenthesis Step 2: Combine like terms Example 12: Perform the indicated operation and simplify (5x 4x + 10) + (3x 2x -12) = 5x -4x + 10 + 3x -2x -12 = 8x 6x + 2 Step 1: Remove the parenthesis Subtracting Polynomials If there is a minus sign in front of the ( ) then distribute it by multiplying every term in the ( ) by a negative one Step 2: Combine like terms Example 13: Perform the indicated operation and simplify (5x 2y + 1) (2x 7y + 4) = 5x 2y + 1 2x + 7y 4 = 3x + 5y 3 8
Multiplying Polynomials When multiplying two polynomials together, use the distributive property on the first polynomial until every term in it has been multiplied times every term in the other polynomial (Monomial) (Monomial) Example 14: Multiply (-7x ) (5x ) = (-7)(5) (x x ) = -35x (Monomial) (Polynomial) Example 15: Multiply -2(5ab + 3a b + 7a b ) = 10a b - 6a b - 14a b (Binomial) (Binomial) Example 16: Multiply (3x + 5) (2x 7) = 6x 21x + 10x 35 = 6x 11x 35 (Polynomial) (Polynomial) Example 17: Multiply (3y -1) (2y + 5y -8) = 6y + 15y 24y - 2y -5y + 8 = 6y + 13y 29y + 8 9
Dividing A Polynomial by A Monomial Step 1: Use the distributive property to write every term of the numerator over the monomial in the denominator. Step 2: Simplify the fractions Example 18: Divide " = + - " = + 3x - " = x + 3x - " 10