degree -6x 3 + 5x 3 Coefficients:

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1 Date P3 Polynomials and Factoring leading coefficient degree constant term coefficients Degree: the largest sum of eponents in a term Polynomial: a n n + a n-1 n a 1 + a 0 where a n 0 E.1 Identify the coefficients & degree of Coefficients: Degree: Perform the operation on the polynomials: E. ( ) ( ) E.3 a) ( + 5)( 6) b) ( 3 6)( 4 + 1) Page 1 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

2 Product of a Sum & Diff. = (a b)(a + b) = E.4 a) (3 )(3 + ) b) 3 3 = c) y 3y = Square of a sum or difference: (a + b) = (a b) Pascal s Δ: Cube of a sum: (a+b) 3 = 1 Row Row Row E.5 ( 3) 3 = GCF (always look for the GCF 1 st ): E.6 a) b) 8w y 4wy Page of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

3 Difference of Two Squares ( terms): E.7 a) 196 b) c) + 9 d) 5 ( 1) Sum & Difference of Two Cubes ( terms): a 3 + b 3 = a 3 b 3 = E.8 a) b) Grouping (4 terms): E.9 a) b) a b a + b Always check to see if you can factor again! E.10 a) b) 7 15 E.11 a) 4 + y 30y b) 4 16 Quadratic Form: E.1 a) b) Page 3 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

4 Date P4 Day 1: Rational Epressions E1. If the numerator and denominator are both polynomials the fractional epression is also a rational epression. A. 5 B What would make the two rational epressions equal? Domain: set of real #s that are defined in an epression by 0 is undefined E. a) 51? 3 b) ? neg # isn t a real # E.3 State the domain of the epression: 7 a) 71 3 b) c) 7 49 {/all real #s, } 3 {/ > -3} Page 4 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

5 1. Factor Reducing Rational Epressions (Restrictions). State restrictions 3. Cancel E.4 a) 4 44 b) c) Reflection: Page 5 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

6 Date P4 Day : Rational Epressions 1. Factor. State restrictions 3. Divide out common factors (NOT common terms!) I. Multiplying & Dividing E.1 a) t t t 6 t 3 6t 9 t 4 b) 3 y 8 y 5y6 3 y 4y It is important to recognize opposite factors: 1 ( 1) E. a) b) 69 9 Page 6 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

7 II. Adding & Subtracting (leave denominator factored) E.3 a) 3 4 b) III. Compound (Comple) Fractions E.4 a) 3 b) E Reflection Page 7 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

8 Date B3 Solving Equations Solving Radical Equations: E.4 a) 7 5 b) c) d) Solving Absolute Value Equations: E.5: Find the solution set for the equation Isolate the absolute value epression. There are two values that have an absolute value of 8, So Solve each equation for. Solutions: You Try: Solve the inequality and graph the solution on a number line Page 8 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

9 Quadratic Equation: an equation of the form a b c 0, with a 0 Methods for Solving a Quadratic Equation Square Roots E.1: Solve the equation 1 5 by etracting square roots. Isolate the epression that is squared: Square root both sides of the equation: Reminder: Solve for : Set Notation: (rationalized) Completing the Square Note: a must equal 1 in order to complete the square. E.: Solve the equation 81 0 by completing the square. Move c to the other side of the equation: Complete the square by adding b to both sides: Rewrite the perfect square trinomial as a binomial squared: Solve by square roots: Page 9 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

10 b b 4ac Quadratic Formula a o Discriminant: b 4ac E.3: Solve the equation using the quadratic formula. Rewrite in standard form and determine a, b, and c. Find the discriminant to determine the number of solutions. b b b 4ac 0 two real solutions 4ac 0 no real solutions ( comple solutions) 4ac 0 one real solution 8 6 Use the quadratic formula to solve. Solution set: Factoring: use the zero product property E.4: Solve the equation by factoring Set the equation equal to zero. Factor the quadratic. (ac method is shown) Split the middle term: Factor by grouping: Set each factor equal to zero and solve: Page 10 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

11 Solving Rational Equations E.5: Find the value(s) of that make the equation true Factor and find the LCD. LCD: Multiply each term by the LCD. Simplify and solve the resulting equation. Page 11 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

12 Solving an Equation Graphically Method 1: Find the zeros (roots). E.6: Solve the equation graphically. Set the equation equal to zero. Graph the function and find the zero(s) y 3 Solutions: Method : Finding the point(s) of intersection. E.7: Solve the equation graphically Graph both sides of the equation as two separate functions. Then find the point(s) of intersection. Solution: Think About It: What does the Y-value represent in the point of intersection? You Try: Solve for in the equation a b c 0 by completing the square. Page 1 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

13 Sample SAT Question(s): Taken from College Board online practice problems. 1. If a 0 and 5 5 a, what is the value of? a (A) 5 (B) 1 (C) 1 (D) (E) 5. If 6, which of the following must be true? (A) 6 (B) 3 (C) 0 (D) (E) Reflection: True or False: An absolute value equation always has two solutions. Eplain your answer. Page 13 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

14 Date: B4 Solving Inequalities Solving a Linear Inequality: E.1: Solve and graph the solution on the number line. Epress the solution in interval notation or Note: When multiplying or dividing by a negative number, you must flip the inequality sign. Interval notation: Solving a Compound Inequality: E.: Solve and graph. Epress in interval notation Solving an Absolute Value Inequality If If a, then a a. Or is between a and a. a, then a or a. E.3: Solve the inequality. 4 1 Since the absolute value is less than 1, the value of the epression inside the absolute value must be between 1 and 1. Solve the compound inequality. *Have students pick a value within the interval to verify that it is a solution to the original inequality. E.4: Solve the inequality and verify your solution graphically. 3 7 Since the absolute value is greater than or equal to 7, the value of the epression inside the absolute value must be less than or equal to 7 or greater than or equal to 7. Page 14 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

15 Solve the compound inequality. We can see that the graph of y 3 is greater than or equal to the graph of y 7 when or 5. Solving Quadratic Inequalities Algebraic Method (sign chart) E.5: Solve the inequality Solve the equation to find the zeros. Make a sign chart using test values between and outside of the zeros. + + The solutions to the inequality are the -values that make the epression positive (greater than zero). Set notation: Graphing Method E.6: Solve the inequality graphically. 5 0 Because the graph is below the -ais y 0 between = and =.851, the solution, written in interval notation is: Page 15 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

16 Projectile Motion: When an object is launched vertically from an initial height of s 0 feet and an initial velocity of v 0 feet per second, then the vertical position s of the object t seconds after it is launched is s 16t v0t s0. E.7: A ball is thrown straight up from ground level with an initial velocity of 59 ft/sec. When will the ball s height above the ground be more than 30 ft? Write the equation of the height of the ball. Write an inequality to model the question. Solve by graphing. The ball s height will be more than ft when seconds. You Try: Solve the quadratic inequality algebraically. Then verify your answer graphically Page 16 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

17 Sample SAT Question(s): Taken from College Board online practice problems. 1. If y 3 and, which of the following represents all the possible values for y? (A) y 7 (B) y 7 (C) y 5 (D) y 5 (E) 5 y 7. The figure above shows the graph of a quadratic function in the y-plane. Of all the points y, on the graph, for what value of is the value of y greatest? 3. At a snack bar, a customer who orders a small soda gets a cup containing c ounces of soda, where 1 c is at least 1 but no more than 1. Which of the following describes all possible values of c? A. B. C. D. E c 1 c 1 1 c c c Reflection: Can a quadratic and/or absolute value inequality have no solutions or one solution? Eplain your answer with an eample. Page 17 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

18 7.1 Solving Systems of Equations Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities. Solution of a System of Linear Equations: the ordered pair y, where the lines intersect A solution can be substituted into both equations to make true statements Special Cases: If the graphs of the equations in a system are parallel (do not intersect), then the system has NO SOLUTION. Systems with no solution are called inconsistent. Systems with solutions are called consistent. If the graphs of the equations in a system are the same line (coincident), then the system has INFINITELY MANY SOLUTIONS. Systems with infinite solutions are called dependent. Note: If the graphs are not coincident and intersect, then the system has EXACTLY ONE SOLUTION. Consistent solution(s) Inconsistent Ø Dependent ( ) Independent (1 or Ø) Systems of equations that are dependent graph lines that are coincident. Page 18 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

19 Solving Linear Systems by Graphing E 1: Graph and determine the point of intersection. 3 y 4 y 6 Graph both lines on the same coordinate plane. The lines intersect at. Solving a System by Graphing (not linear) E : Graph and find the solution of the system. 3y 6 y ln 1 Graph both functions on the same coordinate plane. The solution of the system is the point at which the functions intersect: Page 19 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

20 Solving a System of Linear Equations by Substitution E 3: Solve using substitution. 3 y 7 6 y 8 Step One: Solve one of the equations for one of the variables (if necessary). Note: You may choose which variable to solve for. Step Two: Substitute the epression from Step One into the other equation of the system and solve for the other variable. Note: Both variables were eliminated, and the resulting equation is an untrue statement. Solving Nonlinear Systems by Substitution E 4: Solve the system. y y 13 1 Step One: Solve one of the equations for one of the variables (if necessary). Step Two: Substitute the epression from Step One into the other equation of the system and solve for the other variable. Note: We solved for instead of substituting the epression for y to avoid a high-degree polynomial equation to solve. Step Three: Substitute the value from Step Two into the equation from Step One and solve for the remaining variable. Step Four: Write your answer as an ordered pair and check in both of the original equations. Reflection: Page 0 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

21 7. Solving Linear Systems by Linear Combinations (Elimination) E 1: Solve the system. 3 y 5 9 5y 3 Step One: Write the two equations in standard form. Step Two: Multiply one or both of the equations by a constant to obtain coefficients that are opposites for one of the variables. Step Three: Add the two equations from Step Two. One of the variable terms should be eliminated. Solve for the remaining variable. Step Four: Substitute the value from Step Three into either one of the original equations to solve for the other variable. Step Five: Write your answer as an ordered pair and check in the original system. Application Problems Systems of Equations E : Three gallons of a miture is 60% water by volume. Determine the number of gallons of water that must be added to bring the miture to 75% water. Assign labels: w = gallons of water; t = total gallons of miture Write a system: Amount (gallons) Value (percent water) Solve the system: Rewrite second equation. Use substitution. Page 1 of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

22 E 3: A machine takes 3 minutes to form a bowl and minutes to form a plate. The material costs $0.5 for a bowl and $0.0 for a plate. How many bowls and plates were made if the machine ran for 8 hours and $44 was spent on materials? Assign labels: b = # of bowls; p = # of plates Write a system: Time (Multiply by 60 to change hours to minutes.) Money Solve the system: Rewrite equations. Use elimination. You Try: y 6 1. Solve the system: y y 6 (Hint: You do not need to substitute for a single variable. You can use an epression.). A car radiator contains 10 quarts of a 30% antifreeze solution. How many quarts will have to be replaced with pure antifreeze if the resulting solution is to be 50% antifreeze? Reflection: Eplain how you would choose the most efficient method for solving a system of equations. Page of 3 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 7

23 7.3 Multivariable Linear Systems Precalculus Notes:Prerequisite Skills for Calculus Syllabus Objective: 8.4 The student will solve application problems involving matri operations with and without technology. Triangular Form of a System: an equivalent form of a system from which the solution is easy to read; all lead coefficients are equal to 1 E 1: Solve the system in triangular form. y z 7 y z 7 z 3 Use back substitution: Write solution as an ordered triple: Gaussian Elimination: transforming a system to triangular form by Select Equations. Eliminate a variable. Replace one of the Equation. Select Equations. Eliminate the same variable. Replace one Equation. Repeat until you have a Triangular Form 3,,1 (one equ. in 3 var., one equ in var., 1 equ in 1 var.) E : Solve the system using Gaussian Elimination. y z 0 Eq. I y z 3 Eq. II y 3z 7 Eq. III Note: Calculator Trick Rref: Note: All variables were eliminated, and the result is a false statement. Therefore, this system has Special Systems: A linear system can have eactly one solution, infinitely many solutions, or no solution. y z 4 E 3: Solve the system. 3 6y 3z 7 y 4z Page 3 of 3 Precalculus Larson Algebra Review for Calculus