Regents Review #1

1. 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. Rational & Irrational Expressions • SUMS AND PRODUCTS • The sum or product of two rational numbers is always rational. • The sum of a rational number and an irrationalnumber is always irrational. • The product of a non-zero rationalnumber and an irrationalnumber is always irrational. • The sum or product of twoirrationalnumbers may be rational or irrational.

3. Polynomials A polynomial is a sum of terms. Each term is separated by either a + or – sign.

4. Polynomials The degree of a polynomial with one variable is the highest power to which the variable is raised.

5. Polynomials A polynomial is written in standard form when the degrees (exponents) are listed from highest to lowest.

6. 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 The expression can also be simplified this way.2(3x – 2) + 2(2x2 – x + 11)

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

8. 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) Expand 3x - 4 9x2 – 12x – 12x + 16 9x2 – 24x + 16 4) Express the area of the rectangle as a simplified polynomial expression. x + 5 2x3 + 6x2 – 19x + 5 2x2 – 4x + 1

9. Factoring Polynomials What does it mean to factor? Create an equivalent expression that is a product ( )( ) Remember to always factor completely. Factor until you cannot factor anymore!

10. Factoring Polynomials “Go to Methods” • Factor out the GCF • AM factoring 3) DOTS Trinomial ax2 + bx + c , a = 1 Binomial difference of two perfect squares

11. Factoring Polynomials When factoring completely, factor until you cannot factor anymore! 1) 2) The factored form of the polynomial expression is equivalent to the standard form of the polynomial expression.

12. Solving Equations What types of equations (in one variable) do we need to know how to solve? • Equations with rational expressions (fractions) • Quadratic Equations 3) Square Root Equations 4) Literal Equations (solving for another variable) Remember: When solving any type of equation, always use properties of equality and check solution(s).

13. Rational Equations (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

14. Rational Equations How do we solve a rational equation with more than one fraction? Option 1: Combine fractions and create a proportion Option 2: Multiply by the LCD (least common denominator) Example: Solve for x.

15. Rational Equations Create a Proportion Multiply by the LCD FOO 5x + 10 = 70

16. Quadratic Equations ax2 + bx + c = 0 Ex: x2 – 5x = -6 1) ax2 + c = 0 Ex: 2x2 – 32 = 0 2x2 = 32 x2 = 16 x = x = 4 or x = {4,-4} x2 – 5x + 6 = 0 (x – 2)(x – 3)= 0 x – 2 = 0 x – 3 = 0 x = 2 x = 3 x = {2,3} Isolate x2 and take the square root. • Set the equation equal to zero. • Factor. • Set each factor equal to zero and solve. • Zero Product Property

17. Quadratic Equations How do we solve a quadratic equation that cannot be factored? Example: Find the roots of x2 – 2x – 5 = 0. Use the quadratic formula: a = 1, b = -2, c = -5

18. Quadratic Equations The 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 _______ x2 – 2x+ 1= 5 + 1 (x – 1)(x – 1) (x – 1)2 = 6

19. Square Root Equations Solve Isolate the Square both sides of the equation to eliminate the

20. Literal Equations When solving literal equations, isolate the indicated variable using properties of equality. y(a + x) = c Factor out the variable that you are solving for.

21. Literal Equations Multiply both sides of the equation by 2 to eliminate the fractional coefficient.

22. Problem Solving We write equations to solve problems. The next few slides are some examples of different types of problems we have solved this year by setting up an equation to model the situation.

23. Consecutive Integer Problems Find two consecutive integers whose sum is -35. x: 1st consecutive integer x + 1: 2nd consecutive integer -18 -17 (-18 + 1) Remember: Consecutive integers count by 1’s Ex: x, x + 1, x + 2, x + 3…. Consecutive odd or even integers count by 2’s Ex: x, x + 2, x + 4, x + 6… Negative integers don’t change anything

24. Coin Problems Joe has $2.50. He has 7 more dimes than nickels. How many of each does he have? .05x + .10(7 + x) = 2.50 or 5x + 10(7 + x) = 250 5x + 70 + 10x = 250 15x + 70 = 250 15x = 180 x = 12 Joe has 12 nickels and 19 dimes. Check: 12 nickels = 60 cents 19 dimes = $1.90 Total: $1. 90 + $0.60 = $2.50 Remember: (Value)(Quantity) = Total value of Coins $ per coin x how many = total $

25. Ratio Problems Donna wants to make 4lbs of trail mix made up of almonds, walnuts and raisins. She wants to mix one part almonds, two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts cost $9 per pound, and raisins cost $5 per pound. Donna has $15 to spend on the trail mix. Determine how many pounds of trail mix she can make. Ratio 1:2:3 Let x = the amount of pounds of almonds Let 2x = the amount of pounds of walnuts Let 3x= the amount of pounds of raisins 1/3 lb of almonds 2/3 lb of walnuts (2)(1/3) 1 lb of raisins (3)(1/3) She can make a total of 2 lbs of trail mix. 12x + 9(2x) + 5(3x) = 15 12x + 18x + 15x = 15 45x = 15 45 45 x = Check: (1/3)($12) = $4 (2/3)($9) = $6 (1)($5) = $5 $4 + $6 + $5 = $15

26. Age Problems Sue is 5 years older than Ann. In 6 years, Sue’s age will be 11 years less than twice Ann’s age then. How old is each person now? Future Sue will be 11 years less than twice Future Ann x + 11= 2(x + 6) – 11 x + 11 = 2(x + 6) – 11 x + 11 = 2x + 12 – 11 x + 11 = 2x + 1 11 = x + 1 10 = x Right now, Ann is 10 years old and Sue is 15 years old. Remember: It is helpful to organize information in a table prior to creating an equation.

27. Area Problems Carlos wants to build a sandbox for his little brother. He is deciding between a square with side lengths that can be represented by x + 3 units and a rectangular sandbox with a length 1 unit more than the side of the square and a width 1 unit less. If the area of the rectangular sandbox is 63 square feet, find the value of x. Draw a picture of the situation. A = lw 63 = (x + 4)(x + 2) x = -11 (reject, cannot have a negative length or width) The value of x is 5 feet. 63 = x2 + 2x + 4x + 8 63 = x2 + 6x + 8 0 = x2 + 6x – 55 0 = (x + 11)(x – 5) x + 11 = 0 x – 5 = 0 x = -11 x = 5 Check Area of rectangle: 63 sq. ft. Length: 5 + 3 + 1 = 9 ft. Width: 5 + 3 – 1 = 7 ft. (7)(9) = 63 Factor AM

28. Area Problems A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway? A = lw 285 = (2x + 12)(2x + 16) 285 = 4x2 + 32x + 24x + 192 285 = 4x2 + 56x + 192 x + 16 + x 0 = 4x2 + 56x – 93 Solve for x Quadratic Formula a = 4 b = 56 c = -93 x + 12 + x x = -15.5 (reject, cannot have a negative width) x = 1.5 The width of the pathway is 1.5 meters.

29. 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 12thBe there!