Thinking Algebraically Nikolas Catalano Algebra 1 In the

Thinking Algebraically Nikolas Catalano Algebra 1

In the upcoming slides, reviewed will be the following properties; - Addition Property (of Equality) Multiplication Property (of Equality) Reflexive Property (of Equality) Symmetric Property (of Equality) Transitive Property (of Equality) Associative Property of Addition Associative Property of Multiplication Communitive Property of Addition Communitive Property of Multiplication Distributive Property Inverse Property of Addition Inverse Property of Multiplication Identity Property of Addition Identity Property of Multiplication - Multiplicative Property of Zero Closure Property of Addition Closure Property of Multiplication Product of Powers Property Power of a Product Property Quotient of Powers Property Power of a Quotient Property Zero Power Property Negative Power Property Zero Product Property Product of Roots Property Quotient of Roots Property

Addition Property (of Equality) If the same number is added to both sides of an equation, the two sides remain equal. That is, if x = y, then x + z = y + z. Example; If 3 -x=7 x, then 3 -x+x=7 x+x Adding an “x” to both sides will show the relationship between the two problems. Add the opposite of the original “x. ” The original “x” was a negative number, so you need to add “x. ”

Multiplication Property (of Equality) In problems with this property, you must basically find the value of “x. ” In doing so, you will have to divide the term with x into the answer Example; 5 x=3 Divide 5 into the 3, or you can multiply 3 by 1/5 and the result will be the same. If 5 x=3, then 1/5 x 5 x= 1/5 x 3

Reflexive Property (of Equality) Basically, a number times something is equal to that number times that something. In ways this is an “equivalence” property because what is being multiplied is always what it is simplified in the answer. In cases with variables, the answer is identical to what is being multiplied Example; 3 m=3 m

Symmetric Property (of Equality) If and when one term is equal to another term, then the other term is equal to the first term. In other words, if a=b, then b=a. Example; If a 2=b 5, then b 5=a 2 The terms may be in different order, but that does not have an effect on their equality

Transitive Property (of Equality) In some instances with 3 terms, they can all “translate” to one another. Meaning that if the first term equals the second term and the second term equals the first term, then the first term will equal the third term Here is an example of the pairing; If a 2=b 5 and b 5=c 3, the a 2=c 3 The expression has all equal terms, so you can then pair off the terms by two, and each of those would be equal as well

Associative property of Addition You will find an equation within parenthesis and outside the parenthesis will be an addition problem to the inner parenthetic term. In other words, because the order in which addition goes does not have an effect on the outcome, you can switch around numbers and find the same result Example: (7+¼)+ ¾ = 7+(¼ + ¾) As you see, you can rearrange the numbers and end up with the same answer because of the fact that it is an addition problem

Associative Property of Multiplication The property which states that for all real numbers a, b, and c, their product is always the same, regardless of their grouping: (a. b). c = a. (b. c) Example; (5 x 6) x 7 = 5 x (6 x 7)

Commutative Property of Addition Here, you can again rearrange the problem, because it is addition, and find the same result. You are “commuting” numbers around, without any effect to the answer Example; ¼+7+¾=¼+¾+7 The arrangement has no effect on the answer

Commutative Property of Multiplication As you know, if you multiply two numbers, the answer will be the same no matter what order you multiply the numbers in. That goes for multiplying with exponents too. Example; a 2 x b 3 = b 3 x a 2 The product will always be the same, regardless of the order of the terms

Distributive Property When using this property, there will always be a number before a parenthesis with numbers or variables inside 4(a+b) *Note, as you distribute 4 to “a” and 3 to “b”, keep in mind the positive sign will remain 4 a+4 b

Inverse Property of Addition Even with exponents, subtracting a term by its identical term affixed with the same exponent will equal zero Example; a 3+-a 3 = 0 In this case, a 3 minus a 3 will result in nothing remaining

Inverse Property of Multiplication If you have a fraction and then multiply the reciprocal of that fraction, the answer will always be one Example; 3/a x a/3 = 1 Why is this? Because your result when multiplying will be 3 a as the numerator and the denominator

Identity Property of Addition Identity property of addition states that the sum of zero and any number or variable is the number or variable itself. Example; 4+0=4 -10 + 0 = -10 The number keeps its “identity”

Identity Property of Multiplication A basic rule; multiplying something by one will always result in that first term. It will keep its “identity” in tact. a 2 x 1 = a 2 Even with an exponent, the term will always remain if multiplied by one

Multiplicative Property of Zero This property is very basic. Multiplying something by zero results in zero as the answer. Examples; 8 x 0=0 7 x 0=0

Closure Property of Addition This property says that if you add together any two numbers from a set, you will get another number from the same set. If the sum is not a number in the set, then the set is not closed under addition. This property involves whole numbers. Example; a+b=c and b+a=c 5+2=7 and 7+2=9

Closure Property of Multiplication This property deals mostly with real numbers. There is not an equation or expression to represent this property, but if a is real and b is real, then ab would be a real number If a=r and b=r, then ab=r *r= real number

Power of a Product Property Within parenthesis are 2 or more numbers or variables. The exponent outside the parenthesis then applies to BOTH number or variable. Example; (ab)3 Here, you see the exponent, 3, precedes “ab. ” With the rule mentioned above, apply the exponent to “a” and “b”. a 3 b 3 You see the exponent applied to both terms, and this is the final answer

Power of a Power Property When multiplying, multiply the powers, then simplify the base (a 5)3 In other words; (a)5 x 3 Simplifying 5 x 3 will result in (a)15

Product of Powers Property When multiplying, add the exponents while retaining the base number. Example; 72 × 7 6 If you know the way exponents are defined, you know that this means: (7 × 7) × (7 × 7 × 7 × 7) If we remove the parentheses, we have the product of eight 7 s, which can be written more simply as: 78

Quotient of a Powers Property When dividing with exponents, always subtract the numerator’s exponent by the denominator’s exponent What you're really doing here is canceling common factors from the numerator and denominator. Example:

Power of a Quotient Property Whenever a fraction is in parenthesis and outside the parenthesis lies an exponent, the exponent should then apply to both the numerator and denominator ( )2 Then later applying the 2 to the numerator and denominator will result as; a 2 b 2

Zero Power Property Any non-zero base to the zero power always equals 1. Since x 0 is 1 for all numbers x other than 0, it would be logical to define that 00 = 1. (4 ab)0 =1

Negative Power Property When the exponent is negative, simply turn it into a fraction by adding “ 1” as the numerator. Then, the exponent will become its absolute value. If necessary, simplify the denominator. You can, however, easily skip the second step in the equation above. And get the final result by placing a one as the numerator while turning the exponent positive.

Zero Product Property As you know, if a multiplication problem equals zero, one of the terms must equal zero. If you multiply something by zero, the answer is zero Example; If a 2(b-1)=0, then either a 2=0 or (b-1)=0 If one of these terms is zero, then the product will be zero since you are multiplying something by zero

Product of Roots Property Here, you have two non perfect squares. Since they cannot be broken down, you must multiply both the 6 and the 15 remaining under the square root sign Example; x Find the prime factorization of both 6 and 15. This would leave 3 · 2 · 3 · 5 still remaining under the square root symbol. Cleaning this up would look like 32 x 10 under the square root symbol

Quotient of Roots Property For all positive real numbers a and b, b ≠ 0: The square root of the quotient is the same as the quotient of the square roots. The root applies to both terms

Graphing Finding solutions for linear equations; Example: Is (-3, -7) a solution point for f(x) = 2 x-1? The answer is yes because you plug in -3 for “x” and -7 for f(x). *F(x) is another word for the “y” value. Finding slope; The equation is Plug the values into each and that is the slope

Graphing cont. Use the point-slope formula to graph You can turn it into slope-intercept form by putting the numbers and variables to the applicable sides.

Factoring GCF We will look at the following problem and simply find what each 3 terms have in common in greatest terms. 5 x 3 – 10 x 2 – 5 x Including the variables, they each have a 5 and an x in common. So, you will put this in distributive form (see slide 12). 5 x(x 2 – 2 x – 1) As you see, we took out the necessary components of the original term and put it into the distributive property

Trinomials - Perfect Square Trinomials - Reverse Foil

Perfect Square Trinomials A PST is when the 1 st & 3 rd terms are squares and the middle term is twice the product of their square roots. 9 x 2 – 30 x + 25 The 1 st and 3 rd terms are perfect squares, so inside parenthesis goes the square roots of both and outside goes the 2 nd power (3 x – 5) 2 And this is most definitely a PST because half of the product of the perfect square’s roots times 2 is the middle term

Reverse FOIL There will be two parenthesis containing the base products of the inner and outer terms, The middle term is the sum of the outer and inner terms multiplied 6 x 2 -17 x+12 Here, 3 x 2 is the 1 st term and 4 x 3 is the 3 rd term. Don’t forget to include the x with the 1 st term. (3 x-4)(2 x-3) You can check by redoing the foil process.

Binomials - Difference of Squares - Sum or Difference of Cubes

Difference of Squares With this factoring style, you will again use the GCF method, but you will also be breaking it down a little further. First, we will find the greatest common factors for both terms; 75 x 4 – 108 y 2 Finding the GCF will result in this 3(25 x 4 – 36 y 2) By using the reverse of foil method by multiplying the outside and inside terms, you will end up with this conjugate 3(5 x 2 – 6 y) (5 x 2 + 6 y)

Sum/Difference of Cubes This will deal with exponents to the 3 rd power. This is tougher than squares because you need to get a middle term as well as a 1 st and 2 nd term. a 3 - b 3 Here we see a simple cubed problem with only 2 variables. The base of the problem is (a-b). The sign remains negative. The (a-b) will go outside a parenthesis containing the leftovers.

Inequalities compare one number to another with signs. There are 4 signs; > (greater than) > (greater than or equal to) < (less than or equal to) Example; 7> -7 (7 is greater than negative 7, TRUE)

Graphing Inequalities Things to remember. The line faces to the left when greater than and to the right when less than indicating the number is beyond the comparison point. The circle where you begin on the graph is closed if the number compared can be equal to and is open if it is greater or less than in general Example; X<3 (x is less than 3)

Parabolas To graph a parabola, you should know the vertex, x and y intercepts, and the axis of symmetry). To find the coordinates of the vortex, the equation of the first coordinate is B is the 2 nd term of the given parabolic equation while a is the 1 st term. This equation will also find the line of symmetry. You will find the second part of the vertex coordinates by plugging in what you find in the given equation. You will find the x-intercept coordinates by setting y=0. When you find the xintercept you can plug that in to find the y-intercept. The parabola will open up if a in the equation is positive

Functions Discrete means you have to “lift your pencil” to continue graphing. Continuous means you can draw the graph in one motion. Range is found by reading the y coordinate of the graph Domain is found by reading the x coordinates of the graph

Quadratic Formula This formula will help you find the x value

Links for Practice • • • http: //www. mathtv. com (Recommended) http: //mathworld. wolfram. com/topics/Algebra. html http: //www. coolmath. com/algebra/Algebra 1/index. html http: //www. quickmath. com/ http: //www. brainpop. com/math/algebra/ http: //nlvm. usu. edu/en/nav/topic_t_2. html http: //sun. cs. lsus. edu/web. Mathematica/rmabry/factortrinomial. jsp http: //www. internet 4 classrooms. com/gateway_algebra. htm