Section 5.0A Factoring Part 1


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1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) ( )( + 7) B. Where did the first term ( a ) of the trinomial come from in each of the problems above? (Use the words like product, sum, First terms, Outer terms, Inner terms, and Last terms) C. Where did the last term (c) of the trinomial come from in each of the problems above? (Use the words like product, sum, First terms, Outer terms, Inner terms, and Last terms) D. Where did the middle term (b) of the trinomial come from in each of the problems above? (Use the words like product, sum, First terms, Outer terms, Inner terms, and Last terms) E. Wrap Up Honors Algebra Page 1
2 II. Think and Discuss A. Factoring Reversing the FOILing process. Factor the following trinomials (i.e., break them apart to be two binomials again) = ( )( ) =. 10 ( )( ) + =. 1 ( )( ) Remember what you just discovered and what we discussed on the previous page (i.e., start with the first two questions and then verify with the last question). + =. ( )( ) = ( )( ) B. Factoring can also be the reverse of distributing (i.e., what is in common with all the terms?) Try the following 15 y 10 y = ( ) Remember what you do when you distribute ( y + ),. + y + = ( ) now do the opposite. C. What happens when both are together (reverse distribution and reverse FOILing)? Which do you do first? Always do reverse distribution first, then the reverse FOILing. Try the following r + r 5 r = ( )= ( )( ). r + r + 1 r = ( )= ( )( ). r 1 r ( )= ( )( = ) Homework: p. P 1 all Honors Algebra Page
3 Section 5.0B Factoring Part Objective: Factoring polynomials with or more terms. I. Work Together A. Multiply the following binomials. ( a)( y + b). ( + 7)( y + ). ( )( + ). ( + 7)( + 5) B. Looking at the binomials you just multiplied, can you figure out a way to factor the following? a + b + ay + by = ( )( ) y + + y = ( )( ). + 1 = ( )( ) = ( )( ) C. Wrap Up Honors Algebra Page
4 II. Think and Discuss A. Factor the following using the grouping method a + 5 a = ( )+( )= ( )+ ( )=( )( ). rs + st + r + t = ( )+( )= ( )+ ( )=( )( ). 5y 10 y + = ( )+( )= ( )+ ( )=( )( ) a + = ( )+( )= ( )+ ( )= ( )( ) = ( )+( )= ( )+ ( )=( )( ) B. What happens if none of the methods I use is able to factor the problem? Then the polynomial is prime. Try the following y Homework: p all and p all Honors Algebra Page
5 Section 5.1 Graphing Quadratics Objectives: Graph Quadratic Functions. Find the ais of symmetry and coordinates of the verte of a parabola.. Model data using a quadratic function. y = 5 I. Think and Discuss A. Quadratic Functions Form a) Quadratic term b) Linear Term c) Constant Term. Graph of a quadratic a) Symmetrical (1) Def () Ais of Symmetry (a) Def b) Verte (1) () () (b) Equation for a parabola: c) Direction of Opening (1) If a is positive, it opens () If a is negative, it opens B. Which points are on the graph of the function y = 5? (1, 7). (, 0). (0, 5). (, 11) C. What do you think the greatest eponent is for a quadratic function? for a linear function? D. Try This Tell whether each function is linear or quadratic? a). y = ( )( ) b). y = ( + ) c). y = ( + 5 ) d). y = ( 5). Find the ais of symmetry, verte, and determine if the verte is a maimum or minimum point for the following: y = Use you calculator to verify your answers to problem. Homework: p all, 19, 7,  all, 1 Honors Algebra Page 5
6 Section 5.1B Modeling Real World Data Modeling Data A. The table shows the average temperature in Gatlinburg, TN, for each month. Plot the points on a graph. Would it be useful to represent this data with a linear model? Eplain. Month Temp Feb() 5 Apr() 7 Jun() Aug() Oct(10) 71 Nov(11) 5 B. Finding Equations to model Quadratic Functions Find an equation to model the data mentioned, using your graphing calculator. (Hint: It is done the same way you did the linear functions, ecept for one thing.). Use the equation you just found to predict the average temperature in September.. How close was it to the actual temp of 1? Homework: p all Honors Algebra Page
7 Section 5. Solving Quadratic Equations by Graphing Objectives: Solve quadratics by graphing I. Solving Quadratic Functions Graphically A. Solution (Root or Zero) Algebraic Definition. Graphical definition (What would happen if the graph didn't cross the ais or just touched it?) B. Solve the given graph: y = 5. C. Solve the following using your graphing calculator. y = + y. = Homework: p. 11 all, 0 all, 9, 5 and p all Honors Algebra Page 7
8 Objective: Solve polynomials Section 5. Solving Quadratics by Factoring I. Work Together A. Solve for each variable in the following equations. = 0 y = 0 5y = 0 Why were these easy to solve? B. Solve for each variable in the following equations. ab = 1 5y = 75 Can I use the same method I did with the problems in section A above? Eplain C. Wrap Up II. Think and Discuss A. The Process... Honors Algebra Page
9 B. Try the following = 0 10 = 0 ( )( ) = 0 ( )( ) = 0 ( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0 = or = = or = + = 0 1 = 0 ( )( ) = 0 ( )( ) = 0 ( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0 = or = = or = + = 0 1 = 0 ( )( ) = 0 ( )( ) = 0 = 0 or ( ) = 0 or ( ) = 0 = 0 or ( ) = 0 or ( ) = 0 = or = or = = or = or = C. What is wrong with the following problems? Find the error; eplain the error in this person's thought process; and correct the problem. 5 = 15 + = 5 5 ( 9)( + 5) = 15 + = ( 9) = 15 or ( + 5) = 15 + = 5 = or = = 0 ( 9)( + ) = 0 ( 9) = 0 or ( + ) = 0 = 9 or = Homework: p all,  all, 7, 95 all, 59 all, 79 Honors Algebra Page 9
10 Section 5.A Square Roots I. General Information A. Definition of Square Roots B. Note: Square roots have more than one root. Eamples has two square roots.. The nonnegative root is called the principal root. a) is asking for the principle root. b) is asking for the opposite of the principle root. c) ± is asking for the both roots. C. Problems Find each root II. Radical Epressions A. Properties a b =. 11a b. a b = ± 19 B. Eamples on Simplifying a. ± y z y z. 0y y 0y 7. 1m n mn III. Rationalizing the Denominator A. Protocol B. Eamples Homework: p all Honors Algebra Page 10
11 Section 5.B Operations with Radical Epressions I. Adding and Subtracting Radicals A. Note: To add radicals they must be like radical epressions. Treat like they are. B. Eamples II. Multiplying Radicals Eamples ( + )( + ). ( + 5 )( + 5 ). ( 1 + )( 1 ). ( )( 5 7 ) III. Rationalizing the Denominator Eamples 5. 5 Homework: p all Honors Algebra Page 11
12 Section 5. Comple Numbers Goals: To simplify radicals containing negative radicands.. To multiply pure imaginary numbers.. To solve quadratics equations that has pure imaginary solutions.. To add, subtract, and multiply comple numbers. I. General Information A. Rene Descartes (00 yrs ago) came up with a way to solve Proposals: i = 1 where i is not a real number.. i = 1 B. Pure Imaginary Numbers: Eamples 1 = and 11 = =. Simplify: i i = and 5 0 =. Simplify: i 1 = and i 5 =. Solve: + 1 = 0 and a + = 7 0 II. Comple Numbers A. Comple Number Form Real part Imaginary part Eamples ( + 7 i) + ( i). (9 i) (1 i). ( + 5 i)( i). ( + i)(5 i) III. Comple Conjugates Eamples: + 7i 7i ( )( ). ( 9 7i)( 9 + 7i) IV. Comple Numbers in the Denominator Eamples Simplify i 5 i. + 5 i + 7i Homework: p all, 7,,, 50, 1, Honors Algebra Page 1
13 Section 5.5 Completing the Square Goals: To solve quadratic equations by completing the square. Work Together FOIL These ( ). ( + 5). ( 7) Making perfect squares + = ( ) = ( + ). 0 + = ( ) Solve the following problems without setting equations equal to zero. =. ( ) = 1. (1 + ) = = = Wrap Up Think and Discuss Completing the square is a method to solve quadratic equations when they do not factor. Method Get constant on one side of the equation.. Factor out the coefficient in front of the.. Make the variable side into a perfect square (reminder what ever you add to one side must be done to the other).. Square root both sides and simplify. Eamples = = = = 0 Homework: p. 51 all, 7, (0)/, 5, 79, 757 Honors Algebra Page 1
14 Section 5. The Quadratic Formula and Discriminant Goals:. To solve quadratic equations using the quadratic formula.. To use the discriminant to determine the nature of the roots of the quadratic equation. Work Together Solve using the Complete the Square Method a + b + c = 0 Can you write a formula to solve for in all quadratics? Wrap Up Think and Discuss II. The Quadratic Formula is a method to solve quadratic equations when they do not factor. A. Formula: If a b c + + = 0, then ( ) ± ( ) ( )( ) ( ) ( ) = where 0 B. Eamples = = = = 0 III. The Discriminant (Determines Nature of the Roots) A. Formula: If a + b + c = 0, then. B. Translation If D > 0, then. If D = 0, then. If D < 0, then C. Eamples Determine the nature of the roots = = 5 Homework: p all, 1,,, 55 all, Honors Algebra Page 1
15 Section 5.B Sum and Product of Roots Goals:. To find the sum and product of the roots of a quadratic equation. 5. To find all possible integral roots of a quadratic equation.. To find a quadratic equation to fit a given condition. Work Together Solve the following quadratics 15 = = = 0 Find the sum and product of each problem s roots. Sum =. Sum =. Sum = Product = Product = Product = Can you make a conclusion? (Do you see a pattern with the original quadratic and the sum and products?) Wrap Up Think and Discuss Sum and Product Theorem: Formula: If the roots of a + b + c = 0 are r 1 and r, then Eamples Find the quadratic given its roots. Roots are and . Roots are 5 and 1 *. One root is 5 + i *. One root is 1+ *Note: a + bi ( a + b c ) is a root iff a bi ( a b c ). Homework: p all Honors Algebra Page 15
16 Section 5.7 Transformations with Quadratic Functions Goals: To graph quadratic equations of the form y = a( h) + k and identify the verte, the ais of symmetry, and the direction of the opening.. To determine the equation of the parabola from given information about the graph. I. Terms A. Verte B. Ais of Symmetry C. Parent Graph: y = II. Dynamics of y = a( h) + k A. What does h do? Graph: y = ( ). Graph: 10 y = ( + ) B. What does k do? Graph: y = +. Graph: 10 y = C. What does a do? Graph: y 10 =. Graph: y = Regraph these two equations but us a negative coefficient D. Wrap Up y = a( h) + k h. k Verte: Ais of Symmetry:. a Honors Algebra Page 1
17 E. Eamples: Name the verte, ais of symmetry and direction of opening. a) y = ( + 11) + b) y = ( ) +. Put the following quadratics into a) y = + + b) y = a( h) + k y = c) y = Graph the following: a) y = ( + ) 1 b) y = = ( 1) + c) y ( ) III. Find the equation of the parabola A. Eamples: (,1) (,) (,)  . Parabola passes through the verte (5, ) and the point (, ). Homework: p all, (91)/, () /, 59 all, 1,, 59, 5, 9, 7 Honors Algebra Page 17
18 Section 5. Graphing and Solving Inequalities Goals: To graph quadratic inequalities.. To solve quadratic inequalities in one variable. I. Graphing Quadratic Inequalities A. Same linear functions. B. Eamples y < ( ) 1 y II. Solving quadratic inequalities A. Graphically: 0 > < B. Algebraically 0 > < 0 Homework: p all, 0, ()/, 7, 5, 7, 71 Honors Algebra Page 1
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