Regents Review #1

1. (10x3)2 5x5 x – 3=2x 5x 7 Regents Review #1 3x3 – 4x2 + 2x – 1 (x – 4)(2x + 5) Expressions & Equations (x – 5) 2 = 25 (4a – 9) – (7a2 + 5a + 9) 4x2 + 8x + 1 = 0

2. Evaluating and WritingAlgebraic Expressions x2 – y (-2)2 – (-5) 4 + 5 9 3) Express the cost of yshirts bought at x dollars each. • Evaluate x2 – ywhen x= -2 and y = -5 • Express “three times the quantity of 4 less than a number” as an expression. 3(n – 4) xy

3. Simplifying Exponential Expressions 1) xy0 2) (2x2y)(4xy3) 3) (2x3y5)4 24(x3)4(y5)4 16x12y20 8x3y4 x(1) x Multiply coefficients and add exponents. Any nonzero number raised to the zero power equals 1. Raise each factor to the power.

4. Simplifying Exponential Expressions 6) 4) 5) Divide coefficients subtract exponents. Simplify numerator and denominator coefficients by dividing by a common factor. Raise the numerator and denominator to the power of the fraction.

5. Simplifying Exponential Expressions 8) 7) Move powers with negative exponents to the other part of the fraction. Rewrite using positive exponents. Simplify coefficients. Subtract exponents (all results appear in the numerator). Move powers with negative exponents to the denominator and rewrite with positive exponents.

6. Simplifying Exponential Expressions When simplifying exponential expressions, remember… • Use exponent rules to simplify. • When dividing, all results appear in the numerator. Change negative exponents to positive by moving them to the other part of the fraction. • No decimals or fractions are allowed in any part of the fraction.

7. Polynomials When adding polynomials, combine like terms. • Represent the perimeter of a rectangle as a simplified polynomial expression if the width is 3x – 2 and the length is 2x2 – x + 11. 2x2 – x + 11 (3x – 2) + (3x – 2) + (2x2 – x + 11) + (2x2 – x + 11) 2x2 + 2x2 + 3x + 3x – x – x – 2 – 2 + 11 + 11 4x2 + 4x + 18 3x – 2 3x – 2 2x2 – x + 11 Can also simplify 2(3x – 2) + 2(2x2 – x + 11)

8. Polynomials When subtracting polynomials, distribute the minus sign before combining like terms. • Subtract 5x2 – 2y from 12x2 – 5y (12x2 – 5y) – (5x2 – 2y) 12x2 – 5y – 5x2 + 2y 12x2 – 5x2– 5y + 2y 7x2 – 3y

9. Polynomials When multiplying polynomials, distribute each term from one set of parentheses to every term in the other set of parentheses. 3) (3x – 4)2 (3x – 4)(3x – 4) 9x2 – 12x – 12x + 16 9x2 – 24x + 16 4) Express the area of the rectangle as a simplified polynomial expression. 2x3 + 6x2 – 19x + 5 x + 5 2x2 – 4x + 1

10. Polynomials When dividing polynomials, each term in the numerator is divided by the monomial that appears in the denominator. 5)

11. Factoring What does it mean to factor? Create an equivalent expression that is a “multiplication problem”. Remember to always factor completely. Factor until you cannot factor anymore!

12. Factoring “Go to Methods” • Factor out the GCF • AM factoring 3) DOTS

13. Factoring What about ax2 + bx + c when a 1? Factor 5n2 + 9n – 2 GCF is 1, Factor by Grouping! Find two numbers whose product = ac and whose sum = b ac = (5)(-2) = -10 b = 9 The numbers are -1 and 10 5n2 + 10n – 1n – 2 Rewrite the polynomial with 4 terms 5n2 + 10n - 1n – 2 Create two groups 5n(n + 2) – 1(n + 2) Factor out the GCF of each group (5n – 1)(n + 2) Write the factors as 2 binomials

14. Factoring When factoring completely, factor until you cannot factor anymore! 1) A polynomial expression is factored completely when all the factors are prime. 2)

15. Solving Equations What types of equations do we need to know how to solve? • Proportions • Quadratic Equations 3) Square Root Equations 4) Literal Equations (solving for another variable)

16. Solving Proportions Always check solution(s) to any equation 5(3x – 2) = 10(x + 3) 15x – 10 = 10x + 30 5x – 10 = 30 5x = 40 x = 8

17. Solving Quadratic Equations 1) x2 = a Example: x2 = 16 Take the square root of both sides x = x = 4 or x = {4,-4} • x2 + bx + c = 0 Example: x2 – 5x = -6 x2 – 5x + 6 = 0 Set all terms equal to zero (x – 2)(x – 3)= 0 Factor x – 2 = 0 x – 3 = 0 Set each factor equal to zero x = 2 x = 3 Solve x = {2,3}

18. Solving Quadratic Equations What happens when a quadratic equation cannot be factored? Example: Find the roots of x2 – 2x – 5 = 0. Use the quadratic formula: a = 1, b = -2, c = -5

19. Solving Quadratic Equations This equation can also be solved by completing the square. Find the roots of x2 – 2x – 5 = 0. x2 – 2x – 5 = 0 x2 – 2x = 5 x2 – 2x _____ = 5 _______ (x – 1)(x – 1) x2 – 2x+ 1 = 5 + 1 (x – 1)2 = 6

20. Solving Square Root Equations Example: Solve

21. Literal Equations When solving literal equations, isolate the indicated variable using inverse operations

22. Now it’s your turn to review on your own! Using the information presented today and the study guide posted on halgebra.org,complete the practice problem set.Regents Review #2 Friday, May 9thBE THERE!