Class Journal - Math 1300 (10446)

What I hear, I forget; what I see, I remember; what I do, I understand. - Kung Fu Tzu (Confucius)

One learns the thing by doing the thing; for though you think you know it,

you have no certainty until you try. - Sophocles

Session	Date	Read & Study Section	Discussion Topics	Practice HW Problems	Other Info
29	12 - 16		The in-class Final Exam will be on Wednesday, December 16 from 11:30 a.m. - 2 p.m. in our regular classroom, and it will be comprehensive. Bring a #2 pencil and a scientific calculator (no cell phone calculators allowed; no sharing of calculators allowed). MyMathLab homework must be completed by Tuesday, December 8. To prepare for the final exam: (1) You should  print out a copy of and work through all of the problems in the Math 1300 Final Exam Review Sheet; click on the following link for a copy of the Review Sheet. http://cms.uhd.edu/qep/algebra (2) You should attend a Final Exam Review Session. Click here for details. http://cms.uhd.edu/qep/algebra/New%20Final%20Reviews%20Flyer%20Fall%202009.mht You may also get additional help for the course in the Math Lab (N925) and from me (S707).
28	12 – 2	10.6 11.1 11.2	Section 10.6/11.1/11.2 More examples were done of solving equations. Note in section 10.6 that the Pythagorean Theorem is used to find the length of an unknown side of a right triangle. Study example 6 on page 620. Note in section 11.2 that a quadratic equation must be in standard form before you apply the quadratic formula. Study example 2 on page 656. Details about the Final Exam Review Sheet, Final Exam Review Sessions and the Final Exam were discussed.	If you attend a Final Exam Review Session, 100 points will be added to your total in MML before your average is computed.
27	11 - 30	10.6 11.1 11.2	Section 10.6 Because of the multiplication property of radicals, we can check that square roots and squaring are inverse operations: ()^2= = a. This fact gives us a strategy to solve an equation of the form =n: we square each side and get x=n^2. Study example 1 on page 616. Section 11.1 The square root property uses the fact that square roots and squaring are inverse operations, but there is a catch: this property includes a plus-minus symbol, +. See page 645. To solve an equation of the form ^2=n, we extract the square root of each side and get x=+. Study examples 1, 2, 3 and 4 in section 11.1. Section 11.2 The quadratic formula can be used to find the solutions of any quadratic equation that is written in standard form ax^2+bx+c=0. See page 655. Study examples 1, 2 and 3 in section 11.2.	Section 10.6 Vocabulary & Readiness Check # 1 – 4 all Exercises # 1 – 15 odd, 23 - 41 odd, 51-56 all Section 11.1 Vocabulary & Readiness Check # 1 – 2 Exercises # 1, 5, 9, 11, 17, 23 Section 11.2 Vocabulary & Readiness Check # 1 – 6 all Exercises # 1, 3, 5, 7, 19, 25
26	11 - 23		Test 3 (Sections 8.1, 8.2, 9.1, 10.1, 10.2, 10.3, 10.4, 10.5) Bring a pencil and a calculator (no cell phone calculators allowed; no sharing of calculators allowed). MyMathLab homework for the test sections must be completed by Sunday, November 22.
25	11 – 18	10.4 10.5	Section 10.4 Let’s review the meaning of the index number of a radical and the radicand of a radical. A square root has index n=2, a cube root has index n=3, a fourth root has index n=4, etc. Intuitively, the index is the number inside of the pocket of the radical. The expression inside of the root symbol is called the radicand. The radicands of the roots are 3, 5, and 2x, respectively. When two roots have the same index number and the same radicand, they are called like radicals. Only like radicals may be added and subtracted. For example, 5√(2)+ 3√(2) = (5+3) √(2)=8√(2). However, √(2)+ √(3) cannot be simplified since these are not like radicals. When radical expressions are multiplied, we use the multiplication rule for radicals: . Study example 4 on page 604. Section 10.5 In the fraction 2/√3, the denominator √3 is not a rational number. However, we can rewrite this fraction as an equivalent fraction with no radical in the denominator – we say that we will rationalize the denominator. In this method, we use two rules: (1) we can multiply the same number to the numerator and the denominator (a/b) = (a∙c)/ (b∙c); (2) a square root times itself simplifies as √a∙√a= √(a^2)=a. Using both rules, we get 2/√3=(2∙√3)/ (√3∙√3)= (2∙√3)/ (√9)= (2∙√3)/ 3. Study example 1 on page 608.	Section 10.4 Vocabulary & Readiness Check # 5 – 14 all Exercises # 1, 2, 3, 5, 7, 9, 17-29 ever other odd, 47, 49, 53, 61, 63 Section 10.5 Vocabulary & Readiness Check # 2 – 4 all Exercises # 1 – 11 odd, 17, 19, 25
24	11 - 16	10.3	Section 10.3 The product rule of radicals can be used to “simplify a radical by factoring.” Intuitively, the goal is to rewrite a given radical expression so that the resulting radical contains a smaller number than the starting radical. For example, 50 is not a perfect square but it is divisible by the perfect square 25. Therefore, we can simplify √(50) by factoring and we get: √(50)=√(25∙2)= √(25)∙√(2)= 5√(2). Note that 25 is the largest perfect square factor of 50. Similarly, x^2 is the largest perfect square factor of x^3 and so √( x^3)=√( x^2∙x)= √( x^2)∙√(x)= x√(x). Study example 3, page 595, and example 4, page 596. The Pythagorean Theorem can be used to find a formula for the distance between two points (x1,y1) and (x2,y2). It is given on page 598. The midpoint between two points (x1,y1) and (x2,y2) is the point located exactly halfway between the two points. The formula is given on page 599.	See below.
23	11 - 11	10.2 10.3	Section 10.2 The rational exponent 1/n is defined as the nth root, that is, , etc. See page 586. The rational exponent m/n involves the two operations of the nth root and the mth power, that is, , etc. See page 587. The laws of exponents apply to all exponents, including rational exponents. There is a summary of the exponent rules on page 589. Carefully study the examples in section 10.2 Section 10.3 The product rule for radicals on page 593 states: and the quotient rule for radicals on page 594 states: . Carefully study the examples in section 10.3.	See below and Section 10.3 Vocabulary & Readiness Check # 1 – 10 all Exercises # 1, 3, 7, 15, 31, 35-43 odd, 47, 59, 59, 61, 76-68 73-77 odd, 83-89 odd, 109
22	11 - 9	9.1 10.1	Problems with compound inequalities that use the word “and” were reviewed from section 9.1. Section 10.1 There are many different kinds of roots: square roots (index n=2), cube roots (index n=3), fourth roots (index n=4), etc. In radical notation, these roots are written as respectively. To check a square root, you square the answer; to check a cube root, you cube the answer, to check a fourth root, you raise the answer to the fourth power, etc. Study pages 578 – 581. For powers of a variable: to find the square root, you divide by 2; to find the cube root, you divide by 3; etc. Therefore √(x^4)= x^2 [divide the exponent by 2] and [divide the exponent by 3]. To find the root of a product, one-by-one find the root of each factor. Then since the cube root of 8 is 2 and the cube root of x^6 is x^2. Similarly, to find the root of a quotient, just find the root of the numerator and denominator separately: √(16/25)= √(16)/ √(25)=4/5. Function formulas that contain roots are studied in example 6, page 582.	See below and Section 10.2 Vocabulary & Readiness Check # 1 – 12 all # 1 – 11 odd
21	11 - 4	8.1 9.1 10.1	Section 8.1 A problem similar to problem 35 on page 507 was discussed: Find an equation of the line passing through the point (3/5, 4/10) and the point (-1/5, 7/10). Section 9.1 The rules to solve an inequality are similar to the rules to solve an equation: the same number may be added/subtracted/multiplied/divided to each side of the inequality. But there is one major difference: if both sides of an inequality are multiplied or divided by a negative number, then you must reverse the direction of the inequality. See the rules on page 143 (addition property) and page 144 (multiplication). Answers to inequality problems may be written in interval notation. See page 142. To solve a compound inequality with the word AND, we solve each inequality separately, and then we find where the two solution sets overlap or intersect to get the solution set to the compound inequality. Study examples 2 and 3 on page 543. A compound inequality with “and” may be written as a 3-part inequality. The inequality “2<4-x<7” is equivalent to “2<4-x and 4-x<7”. Your goal is to use algebra to isolate x in the middle to get the solution. Study examples 4 and 5 on page 544. Section 10.1 Note that the powers of x that are perfect squares are: (x)^2=x^2, (x^2)^2=x^4, (x^3)^2=x^6, etc. That is, the even powers of x are perfect squares. Also note that √(x^2 )=x, √(x^4)= x^2 and √(x^6)= x^3, i.e., you divide the exponent by 2 to get the square root. To simplify the square root of a product, you can distribute the square root: √(25x^4)= √(25) √(x^4)=5x^2. Study the examples in section 10.1 Class Activity 9 was done in class today.	Section 9.1 Vocabulary & Readiness Check # 1, 7, 8 Exercises # 13, 15, 17, 19, 21, 25, 27, 45, 49 Section 10.1 Vocabulary & Readiness Check # 1 – 12 all # 1 – 11 odd, 19 – 53 odd, 57 – 71 odd, 79
20	11 - 2	8.1 8.2	Section 8.1 Perpendicular lines: when the slopes are multiplied together, you get -1. For example the lines y=2x+6 and y=(-1/2)x+3 Are perpendicular, since the product of the slopes is (2)(-1/2)=-1. For perpendicular lines, it is also said that one slope is the negative reciprocal of the other slope. For example, if one line has slope ¾, the negative reciprocal of ¾ equals (-4/3). See example 7 on page 505. The standard form of the equation of a line is ax+by=c where both the x- and y-terms are written on side of the equation and only a number on the other side. The equation 2x+3y=-6 is written in standard form. See example 6 on page 505. Section 8.2 Given a function formula f(x)=2x+1, to calculate f(3), we substitute 3 for x on both sides of the equation: f(3)=2(3)+1=6+1=7. To graph a function, we let y=f(x). For example, since f(3)=7, this means the point (x,y)=(3,7) is a point on the graph of this function. This process can also be done in reverse: given the graph of a function f(x), we can find the value of f(4) by inspecting the graph to see if there is a point on the graph that has x=4. In example 1, page 510, the point (4,2) is on the graph so we write f(4)=2. Also, we can find all x-values such that f(x)=1, by looking for any points on the graph that have y=1. Carefully study example 1 on page 510. The notation √a stands for the positive or principal square root of a. To check a square answer, you square: √36 = 6 because 6^2=36. Study example 4 on page 513.	See below and Section 8.2 Vocabulary & Readiness Check # 3 – 4 # 1 – 7 all, 13, 14, 17-27 odd
19	10 - 28	8.1	Section 8.1 Function Notation. For the function f(x)=2x+6, we can substitute a number for x and get only one number as the answer. For example, to evaluate f(1), we must substitute 3 for x on both sides of the function equation and we get f(1)=2(1)+6=2+6=8, which is read as “f of 1 equals 8.” The function f(x)=2x+6 can be written as y=2x+6. This means that a point (x,y) is contained on the graph of f(x) only if y=f(x). For example, (x,y)=(1,8) is contained on the graph since y=8=f(1). Let’s find the equation of the line that goes through the two points (1,2) and (5,6). One way to do this is to use the slope formula twice. (1) Calculate the slope: ; (2) Use the slope formula again, but substitute 1 written as a fraction for the slope, and use only one given point, say (1,2), and use (x,y) to stand for any other point on the line. You get: . This is a proportion and so the cross-products are equal: (y-2)(1)=(x-1)(1). Next, simplify and isolate y on one side. You get y=x+1 which is the slope-intercept form of the line, this can also be written using function notation as f(x)=x+1. An alternate solution method is used in example 4 in the book on page 504. Parallel lines have equal slopes. We practiced an example in class like exercise 41: write the equation of the line that passes through the point (3,8) and that is parallel to f(x)=4x-2. Class Activity 8 was done in class today.	See below.
18	10 - 26	8.1	Section 8.1 (1) There is exactly one line that goes through the two given points (1,2) and (5,6); draw it. (2) There is exactly one line that goes through the given point (1,2) that has slope equal to 3/4 = rise/run; draw it. The slope of the line that goes through the points (x1,y1) and (x2,y2) equals (y2 - y1)/ (x2 - x1) = rise/run. A rising line has positive slope; a falling line has negative slope; a horizontal line has zero slope; a vertical line has undefined slope. The graph of the equaion y=mx+b or f(x)=mx+b is a straight line that goes through the point on the y-axis (0,b) which is called the y-intercept and that has slope=m. So the equation y=mx+b or f(x)=mx+b is called the slope-intercept form of a line. Here are examples from 8.1 that you should study and learn to do: Example 1: Graph g(x)=2x+1. Example 3: Find (write) the equation of the line with slope -3 and y-intercept (0,-5). Example 4: Find (write) the equation of the line that goes through the points (4,0) and (-4,-5).	Section 8.1 Vocabulary & Readiness Check # 1 – 10 VRC # 1 – 13 odd, 17 – 23 odd, 27, 29, 33 – 43 odd, 56
17	10 – 21		Test 2 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7) Bring a pencil and a calculator (no cell phone calculators allowed; no sharing of calculators allowed). MyMathLab homework for the test sections must be completed by Tuesday, October 20.		To help prepare for the test, you should work through the Chapter 7 Test Prep Video that is in the book on page 499 and that is also in available in MyMathLab.
16	10 – 19	7.5 7.6 7.7	Section 7.5 Problem: solve an equation containing a fraction. Solution: 1^st, clear the equation of fractions by multiplying both sides by the LCD of all equations in the equation. Study example 4 on page 467. Section 7.6 In a proportion, the cross products are always equal. When two triangles, this means corresponding angles are equal, and corresponding sides are proportional. Study example 4 on page 475. Section 7.7 To simplify a complex fraction, it is helpful to first rewrite it by using the rule .	See below.
15	10 - 14	7.5 7.6 7.7	Section 7.5 Many formulas in math, business and science have more than one variable. You can use algebra to solve the formula equation for a specified variable. Study example 7 on page 468. Section 7.6 A proportion is a mathematical statement (equation) that 2 ratios (fractions) are equal. For example, the proportion (1/2)=(4/8) can be read as “1 is to 2 as 4 is to 8.” In a proportion, the cross products are always equal; see page 472. This rule is useful to solve proportion equations. Study examples 1 and 2 on page 473. Section 7.7 A complex fraction is a fraction whose numerator contains a fraction, or whose denominator contains a fraction. To “simplify a complex fraction” means to rewrite it as a simple fraction. A useful rule that is often used to rewrite a complex fraction is: . Class Activity 7 was done in class today.	Section 7.6 Solve each proportion # 1 – 7 odd Use a proportion to solve each problem # 9 - 19 odd, 29, 31, 55, 56, 67, 69 Section 7.7 Vocabulary & Readiness Check # 1 – 2 VRC Simplify the complex fraction # 1 – 13 odd, 17, 19, 29 – 31 odd
14	10 - 12	7.2 7.4 7.5	Section 7.2 Before you multiply or divide fractions, it is recommended that you first factor each numerator and denominator. Study example 3 on page 445. Section 7.4 Two fractions can be added or subtracted by multiplying the denominators to get the new denominator and cross-multiplying to get the new numerator: and . Section 7.5 To solve equations containing fractions (rational expressions), you may clear the equation of fractions by multiplying each side of the equation by the LCD. Carefully study each example in section 7.5.	Section 7.5 Solve each equation and check each solution # 1 – 23 odd, 25 – 41 every other odd Solve for the indicated variable # 43 – 51 odd
13	10 - 7	7.3 7.4	Section 7.3 To add/subtract fractions with like denominators, you add/subtract the numerators and divide by the common denominator. Equivalent fractions are fractions that are equal after each is reduced to lowest terms. To build up a fraction to an equivalent fraction, you must multiply the numerator and the denominator by the same number: (a/b) = (a∙c)/ (b∙c). For example, 3/4=(3∙x)/(4∙x)=(3x)/(4x). The steps to find the least common denominator are on page 452. Section 7.4 To add fractions with unlike denominators: 1^st, find the LCD; 2^nd, build up each fraction to have the LCD; 3^rd, the fractions now have a like denominator and may be added as before. Carefully study the examples in section 7.4. Class Activity 6 was done in class today.	Section 7.4 Vocabulary & Readiness Check # 1-4 Perform the indicated operation # 1-57 every other odd, # 59-67 odd, Perform the indicated operations # 75, 77
12	10 - 5	7.2 7.3	Section 7.2 Review the important rules: (1) only common factors can be divided out from a fraction, that is, (a∙c)/(b∙c)=(a/b); (2) to multiply fractions, multiply the numerators and then multiply the denominators, that is, (a/c)∙(b/d)=(ab)/(cd), then check if the fraction can be reduced; (3) to divide fractions, multiply the first fraction by the reciprocal of the second fraction, that is, (a/c)(b/d)=(a/c)∙(d/b)=(ad/bc) then check if the fraction can be reduced. Section 7.3 To add or subtract fractions, first check if the fractions have like or unlike denominators. To add fractions with like denominators, keep the common denominator and add the numerators: a/c + b/c = (a+b)/c. To subtract fractions with like denominators, keep the common denominator and subtract all of the second numerator from the first numerator: a/c - b/c = [(a)-(b)]/c. This means that if the second numerator contains terms, then the “-” sign must be distributed to each term in the second numerator. Carefully study the examples in section 7.3.	Section 7.3 Vocabulary & Readiness Check # 1-5 Perform the indicated operation and simplify the result, if possible. # 1 – 47 every other odd, 53-67 all
11	9 - 30	7.1 7.2	Section 7.2 The fundamental rule to multiply fractions is: (a/c)∙(b/d)=(ab)/(cd). “multiply numerators and multiply denominators” Study the multiplication examples: examples 1, 2, and 3. The reciprocal of the fraction a/b is b/a, that is, you invert the fraction. The fundamental rule to divide fractions is: (a/c)(b/d)=(a/c)∙(d/b). “multiply by the reciprocal” Study the division examples: examples 4, 5, 6, 7, and 8. Class Activity 5 was done in class today.	Section 7.2 Vocabulary & Readiness Check # 1-2 Find each product and simplify # 1-15 odd Find each quotient and simplify # 17 – 29 odd Perform the indicated operation # 31 – 47 odd, 51, 53, 73, 75
10	9 - 28	7.1	Section 7.1 A fraction is in lowest terms if the numerator and denominator have no common factors other than 1 or -1. To simplify a fraction or a rational expression means to write it in lowest terms. The fundamental rule to simplify a fraction is: (ac)/(bc)=(a/b)(c/c)=(a/b)(1)=(a/b). In words, this says that “factors of 1” can be removed, or that a common factor in the numerator and denominator can be divided out from a fraction. However, a common term in the numerator and denominator cannot be divided out. Study the Helpful Hint box on page 437. Carefully study each example in section 7.1. Function Notation An algebraic formula can be used as the rule for a function. The expression f(x) is read as “f of x.” If f(x)=3x+1 and we are asked to calculate f(0), this means we should substitute 0 for x all the way across this equation to get the answer. Therefore, f(0)=3(0)+1=0+1=1.	Section 7.1 Vocabulary & Readiness Check # 1-2, 4-12 Simplify each rational expression # 17-39 odd, 45, 77 Find each function value # 51, 53
9	9 - 23		Test 1 (Sections 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7) Bring a pencil and a calculator (no cell phone calculators allowed; no sharing of calculators allowed). MyMathLab homework for the test sections must be completed by Tuesday, September 22.		To help prepare for the test, you should work through the Chapter 6 Test Prep Video that is in the book on page 430 and is also in available in MyMathLab.
8	9 – 21		Problems related to test 1 topics were discussed.	See below.
7	9 - 16	6.6 6.7	Section 6.6 A quadratic equation can be written in the standard form ax^2+bx+c=0 where the coefficient a can be any number except 0; note that the highest power of the variable x is 2 in a quadratic equation. A quadratic equation must be solved by using the Zero Factor Theorem, that is, by rewriting one side of the equation as a product and having the other side to be zero. Study example 3: Solve x^2 – 9x – 22 = 0. Solution: Factor the left side to get the form “product = 0”, then apply the zero factor theorem. x^2 – 9x – 22 = 0 (x+2)(x-11) = 0 x+2=0 Or x-11=0 x=-2, x=11 are the solutions. These can be checked by substitution. Study example 5: Solve x(2x – 7) = 4. Solution: 1^st, write the equation in standard form. x(2x – 7) = 4 2x^2 – 7x – 4 = 0 Next, factor the left side to get the form “product = 0”, then apply the zero factor theorem. 2x^2 – 7x – 4 = 0 (2x+1)(x-4) = 0 2x+1=0 Or x-4=0 x=-1/2, x=4 are the solutions. These can be checked by substitution. Study all the examples in section 6.6. Section 6.7 Tips to solve a stated problem: (1) Understand – read and reread the problem; (2) Translate the words into math symbols; (3) Solve the math problem; (4) Interpret – answer the question in the problem. It is important to review geometric formulas: perimeter and area of a triangle, and perimeter and area of a rectangle. Translating words takes practice too: see page 418. Study every example in section 6.7. Class Activity 4 was done in class today.	See below and Section 6.7 Geometric figure problems # 1, 3, 5, 7, 8, 9, 10, 12, 15, 16, 33, 39, 40, 61 Given formula # 13, 14, 35, 41, 42 Number problems # 21, 23
6	9 - 14	6.5 6.6	Section 6.5 The sum of two cubes can be factored. a^3 + b^3 = (a)^3 + (b)^3 = (1^st + 2^nd) (1^st squared – 1^st × 2^nd + 2^nd squared) = (a+b)( a^2 – ab + b^2) where 1^st means the first base and 2^nd means the 2^nd base. Note the sign in the first pair of parentheses is the same as the sign in the starting expression, but in the second pair of parentheses, the sign of the second term is opposite of the sign in the first pair of parentheses. The difference of two cubes can be factored using a similar pattern. a^3 – b^3 = (a)^3 – (b)^3 = (1^st – 2^nd) (1^st squared + 1^st × 2^nd + 2^nd squared) = (a–b)( a^2 + ab + b^2) where 1^st means the first base and 2^nd means the 2^nd base. Note the sign in the first pair of parentheses is the same as the sign in the starting expression, but in the second pair of parentheses, the sign of the second term is opposite of the sign in the first pair of parentheses. Study all of the examples in section 6.5. Section 6.6 The Zero Factor Theorem says that if a product of real numbers equals zero, then at least one of the factors equals zero, that is, if ab=0 then a=0 or b=0. See page 406. The zero factor theorem can be used to help solve equations. Study example 1: solve (x-3)(x+1)=0. Solution: Apply the zero factor theorem: either x-3=0 or x+1=0. Solving each of these smaller linear equations gives the two solutions x=3 and x=-1. The zero factor theorem can be used to solve an equation of the form “product=0” even if there are more than 2 factors in the product. Study example 8: solve (5x-1)(2x^2+15x+18)=0. Solution: Apply the zero factor theorem: either 5x-1=0 or (2x^2+15x+18)=0. The equation 5x-1=0 has solution x=1/5. The trinomial can be factored and gives two more solutions. Altogether then there are three solutions, x=1/5, x=-3/2, x=-6.	See below.	All homework assignments for chapter 6 (6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, and 6.7) in MyMathLab must be completed by Tuesday, September 22, which is the day before Test 1.
5	9 - 9	6.3 6.4 6.5	Section 6.3/6.4 To factor a trinomial in 2 variables, you still focus on the coefficients in the trinomial. For example: Factor 3x^2 + 11xy + 6y^2. Solution: The key number here is (3)(6)=18. Then we look for factors of 18 whose sum is 11, which is the coefficient of the middle term. The factors are 2 and 9 since (2)(9)=18 and 2+9=11. Next we substitute two terms 2xy+9xy for the middle term 11xy, and finally we factor by grouping. 3x^2 + 11xy + 6y^2 =3x^2 + 9xy +2xy + 6y^2 =(3x^2 + 9xy) + (2xy + 6y^2) = 3x(x+3y)+2y(x+3y) = (x+3y)(3x+2y) is the final answer. Section 6.5 The difference of two squares can be factored: a^2 - b^2 = (a+b)(a–b) = (a–b)(a+b). But the sum of two squares cannot be factored using integers: x^2+1 is prime. Also, the sum of two cubes can be factored. a^3 + b^3 = (a)^3 + (b)^3 = (1^st + 2^nd) (1^st squared – 1^st × 2^nd + 2^nd squared) = (a+b)( a^2 – ab + b^2) where 1^st means the first base and 2^nd means the 2^nd base. Also note that in the second pair of parentheses, the sign of the second term is opposite of the sign in the first pair of parentheses. Study all of the examples in section 6.5. Class Activity 3 was done in class today.	See below and Section 6.6 Vocabulary & Readiness Check # 1-10 Solve each equation # 1, 5, 9, 19, 23, 25, 27, 37, 49, 51, 55, 59, 65, 67, Find the x-intercepts # 77, 79, 83	There are homework assignments for 6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, and 6.7 in MyMathLab.
4	9 - 2	6.3 6.4 6.5	Section 6.3/6.4 To factor a trinomial ax^2 + bx + c, you can use the Key Number Method; see the link to the steps in the last column. For example, let’s use the key number method to solve example 1, page 382: Factor 3x^2 + 11x + 6. Solution: The key number here is (3)(6)=18. Then we look for factors of 18 whose sum is 11, which is the coefficient of the middle term. The factors are 2 and 9 since (2)(9)=18 and 2+9=11. Next we substitute two terms 2x+9x for the middle term 11x, and finally we factor by grouping. 3x^2 + 11x + 6 =3x^2 + 9x +2x + 6 =(3x^2 + 9x) + (2x + 6) = 3x(x+3)+2(x+3) = (x+3)(3x+2) is the final answer. Remember to always check for the GCF first. Study example 11 on page 387: Factor 162x^3 - 144x^2 + 32x. Section 6.5 The difference of two squares can be factored: a^2 - b^2 = (a+b)(a–b) = (a–b)(a+b). Study example 1: Factor x^2 – 25. Solution: x^2 – 25 = (x)^2 – (5)^2 = (x+5)(x-5). Study example 2b: Factor 25a^2 – 9b^2. Solution: 25a^2 – 9b^2 = (5a)^2 – (3b)^2 = (5a+3b)(5a-3b).	See below and Section 6.4 Factor by grouping # 1 – 11 odd, 15, 17, 27, 43, 46 Section 6.5 Vocabulary & Readiness Check # 2, 4-7 Exercises Factor completely # 1-9 odd, 15-21 odd, 37, 41, 43, 49, 61, 63, 81	Click on this link to see another method for factoring a trinomial: Factoring Trinomials Using the Key Number Method.
3	8 - 31	6.2 6.3	Section 6.2 To “factor completely” an expression means to keep factoring until you cannot factor anymore. In any factoring problem, you should first factor out the greatest common factor. Study Example 9: Completely Factor 3m^2 – 24m – 60. Solution: The GCF here is 3. Then 3m^2 – 24m – 60 = 3(m^2 – 8m – 20) = 3(m+2)(m-10). Also, if the first term is negative, you should include -1 in the GCF. Study Exercise 65: Completely Factor -x^2 + 12x – 11. Solution: The GCF here is -1. Then -x^2 + 12x – 11 = -1(x^2 – 12x + 11) = -1(x-11)(x-1). Section 6.3 To factor a trinomial ax^2 + bx + c is similar to the strategy in section 6.2. Study Example 2: Factor 3x^2 + 11x + 6. Solution: 3x^2 + 11x + 6 = (3x+1)(x+6). Study the examples in section 6.3. Class Activity 2 was done in class today.	See below and Section 6.3 (page 387) Vocabulary & Readiness Check # 2 – 20 Exercises Complete each factored form # 1, 3, 5 Factor completely # 7 – 13 odd, 17, 19, 23, 25, 35, 39, 41, 47, 55, 57, 59, 63, 65, 75, 81
2	8 - 26	6.1 6.2	Section 6.1 “To factor” an algebraic expression means to rewrite the expression as a product, if possible. This can be done in two steps: (1) find the GCF; (2) factor out (divide out) the GCF from each term in the starting expression. If the expression cannot be factored, it is prime. To check a factoring problem, multiply out the answer and it should equal the original problem. Study Example 4: Factor y^5 – y^4. Solution: The GCF here is y^4. Then y^5 – y^4= y^4 ∙y – y^4 ∙1= y^4(y – 1). Study Example 5: Factor -9a^5 + 18a^2 – 3a. The GCF here is -3a. Then -9a^5 + 18a^2 – 3a = -3a(3a^4 - 6a + 1). Study Example 9: Factor 5(x+3) + y(x+3). Solution: The GCF here is (x+3). Then 5(x+3) + y(x+3) = 5∙(x+3) + y∙(x+3) = (x+3)(5+y). Note: If the first term has a negative number coefficient, then the GCF includes -1. The steps “to factor by grouping” are on page 371. Study Example 13: Factor by grouping 2a^2 + 5ab + 2a + 5b. Solution: 2a^2 + 5ab + 2a + 5b = (2a^2 + 5ab) + (2a + 5b) = a(2a + 5b) + 1(2a+5b) = (2a + 5b)(a + 1). Section 6.2 The factored form of a trinomial x^2 + bx + c is (x+?)(x+?) where the product of the missing numbers is c and the sum of the missing numbers is b. See page 376. (1) If b and c are both positive, then the missing numbers are both positive. Study example 1, factor x^2 + 7x + 12. (2) If b and c are both negative, then one of the missing numbers is positive and one is negative. Study example 4, factor r^2 – r – 42. (3) If c is positive and b is negative, then both missing numbers are negative. Study example 2, factor x^2 – 12x + 35.	See below and Section 6.2 (page 379) Vocabulary & Readiness Check # 1 – 10 Exercises Factor completely each trinomial # 1 – 13 odd, 19, 21, 23, 27 – 55 EOO, 65
1	8 - 24	6.1	Section 6.1 Since 6=2*3, that is, 6 is the product of 2 and 3, the numbers 2 and 3 are called factors of 6. For any list of algebraic expressions, our objective is to find the greatest common factor (GCF): 1^st, factor each expression as a product of primes; 2^nd, identify the common prime factors; 3^rd, the product of all the common factors is the GCF. See example 1, page 368, the GCF of 15, 18 and 66 is 3. The GCF of a list of powers of the same variable is the smallest power of the variable in the list. See example 2, page 368, the GCF of x^2, x^3, and x^5, is the smallest power x^2 in the list. Note in example 3, page 369, that a negative number is written so that -1 is a factor: -18y^2=-1*2*3*3*y^2.	Section 6.1 (page 373) Vocabulary & Readiness Check # 1 – 12 all Find the GCF # 1- 21 EOO, 23 – 37 odd Factor a GCF with a negative coefficient # 49, 51, 53 Factor each polynomial by grouping # 55, 57, 59, 61, 67, 69, 71, 73 Factor # 75, 79, 83, 85 Review and Preview # 92, 93, 96	You must register in MyMathLab during the first two weeks of the semester (by September 6). Class Activity # 1: Register in MyMathLab and in the UHD Math Portal during the first week of classes (by Aug. 30).