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### Expressions and Formulas

- When evaluating an expression, follow the standard order of operations: first calculate exponents, then multiply and divide from left to right, then add and subtract from left to right. BUT: parentheses always override the standard rules. Always work within parentheses first.
- Review and understand the key terms: variable, expression, monomial, constant, coefficient, polynomial, term, like terms, binomial, trinomial.

### Expressions and Formulas

^{3}− 2x

^{2}− xy when x = − 1 and y = − 2

- Subsitute x = − 1 and y = − 2 into the expression. Use parenthesis.
- ( − 1)
^{3}− 2( − 1)^{2}− ( − 1)( − 2) - Follow the order of operations. Exponents First.
- − 1 − 2(1) − ( − 1)( − 2)
- Followed by multiplication
- − 1 − 2 − 2
- Add the three negative numbers to find your solution.

^{2}+ 3x − 3)/(y

^{2}− 2z)] when x = − 2, y = 3 and z = 1

- Substitute x = − 2, y = 3 and z = 1 into the equation. Remember, always use parehtnesis
- [(( − 2)
^{2}+ 3( − 2) − 3)/(3^{2}− 2(1))] - Now do exponents first. Followed by multiplication.
- [(4 + 3( − 2) − 3)/(9 − 2(1))]
- Now multiply any necessary terms.
- [(4 − 6 − 3)/(9 − 2)]
- Simplify the numerator and denominator

^{2}+ 3x + 3 as monomial, binomial or trinomial and state the power of each term.

- The polynomial 3x
^{2}+ 3x + 3 is a trinomial because it has 3 terms. - The power of the first term = 2
- The power of the second term = 1

^{0}= 3.

^{2}y

^{3}z

^{5}as monomial, binomial or trinomial and state the power of each term.

- The polynomial − [3/5]x
^{2}y^{3}z^{5}is a monomial because it only has one term.

^{2}− 25 as monomial, binomial or trinomial and state the power of each term.

- The polynomial x
^{2}− 25 is a binomial because there are 2 terms. - The power of the first term = 2.

^{2}+ 64t tells you the height of the ball from the air after t seconds. How far is the ball from the ground after 2 seconds?

- Given the equation for height, substitute t = 2 for t.
- h = − 16(2)
^{2}+ 64(2) = − 16(4) + 128 = − 64 + 128 = 64

^{2}− 1]

^{2}

- We begin by taking care of the parenthesis (5 − 2) resulting in 3
- [3 + 2(3)
^{2}− 1]^{2} - Next, take care of the exponents inside the brackets.
- [3 + 2*9 − 1]
^{2} - Next, multiply inside the parenthesis
- [3 + 18 − 1]
^{2} - Next, add from left to right
- [20]
^{2} - Lastly, raise 20 to the second power

^{2}) − 2)

- Here there several nested parenthesis. To get started, simplify the inner parenthesis (3 − 2) and work your way outwards.
- 3(2 + (3 − (2)
^{2}) − 2) - Get rid of the exponent
- 3(2 + (3 − 4) − 2)
- Evaluate the inner parenthesis once again
- 3(2 − 1 − 2)
- Evaluate the last parenthesis.

^{5}+ x

^{3}+ 1 when x = − 1 . What do you notice about the result?

- Subsitute x = − 1 into the polynomial
- ( − 1)
^{5}+ ( − 1)^{3}+ 1 = ( − 1) + ( − 1) + 1 = − 1

^{4}+ x

^{2}+ 1 when x = − 1 . What do you notice about the result?

- Subsitute x = − 1 into the polynomial.
- ( − 1)
^{4}+ ( − 1)^{2}+ 1 = (1) + (1) + 1 = 3

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Expressions and Formulas

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Order of Operations
- Variable
- Algebraic Expression
- Term
- Example: Algebraic Expression
- Evaluate Inside Grouping Symbols
- Evaluate Powers
- Multiply/Divide Left to Right
- Add/Subtract Left to Right
- Monomials
- Polynomials
- Formulas
- Example 1: Evaluate the Algebraic Expression
- Example 2: Evaluate the Algebraic Expression
- Example 3: Area of a Triangle
- Example 4: Fahrenheit to Celsius

- Intro 0:00
- Order of Operations 0:19
- Variable
- Algebraic Expression
- Term
- Example: Algebraic Expression
- Evaluate Inside Grouping Symbols
- Evaluate Powers
- Multiply/Divide Left to Right
- Add/Subtract Left to Right
- Monomials 4:40
- Examples of Monomials
- Constant
- Coefficient
- Degree
- Power
- Polynomials 8:02
- Examples of Polynomials
- Binomials, Trinomials, Monomials
- Term
- Like Terms
- Formulas 11:00
- Example: Pythagorean Theorem
- Example 1: Evaluate the Algebraic Expression 11:50
- Example 2: Evaluate the Algebraic Expression 14:38
- Example 3: Area of a Triangle 19:11
- Example 4: Fahrenheit to Celsius 20:41

### Algebra 2

### Transcription: Expressions and Formulas

*Welcome to Educator.com.*0000

*Today is our first lesson for the Algebra II series, and we are going to start out with some review of concepts from Algebra I.*0002

*If you need more detail about any of these concepts, please check out the Algebra I series here at Educator.*0008

*The first session is on expressions and formulas.*0014

*Recall the earlier concepts of variables and algebraic expressions:*0020

*starting out with some definitions, a variable is a letter or symbol that is used to represent an unknown number.*0025

*It could be any letter; frequently, x, y, and z are used, but you could choose n or s or w.*0034

* Algebraic expressions means that terms using both variables and numbers are combined using arithmetic operations.*0045

*Remember that a term is a number, or a variable, or both.*0057

*So, a term could be 4--that is a constant, and it is a term; it could be 2x; it could be y ^{2}.*0065

*And when these are combined using arithmetic operations, then they are known as expressions.*0076

*And when variables are involved, then they are algebraic expressions.*0082

*For example, 4x ^{3}+2xy-1 would be an example of an algebraic expression.*0085

*The rules specifying order of operations are very important; and they are used in order to evaluate algebraic expressions.*0097

*Recall the procedure to evaluate an expression using the order of operations.*0107

*First, evaluate expressions that are inside grouping symbols: examples would be parentheses, braces, and brackets.*0116

*The next thing, when you are evaluating an algebraic expression, is to evaluate powers.*0151

*So, if a term is raised to a power (such as 4 ^{2} or 3^{4}), you need to evaluate that next; that is the second step.*0160

*Next is to multiply and divide, going from left to right.*0175

*You start out at the left side of an expression; and if you hit something that needs to be divided, you do that.*0193

*And you proceed towards the right; if you see something that needs to be multiplied, you do that.*0198

*It is not "multiply all the way, and then go back and divide"; it is "start at the left; any multiplication or division--do it."*0203

*Move to the next step; move towards the right; multiply or divide...and so on, until all of that has been taken care of.*0208

*Finally, you do the same thing with addition and subtraction: you add and subtract from left to right.*0216

*And we will be illustrating these concepts in the examples.*0226

*One thing to recall is that a fraction bar can function as a grouping symbol.*0231

*For example, if I have something like 3x-2x+2, all over 4(x+3)+3, I would treat this as a grouping symbol.*0236

*And I would simplify this as far as I could, going through my four steps; and then I would simplify this;*0248

*and then I would divide this simplified expression on the top by the simplified expression on the bottom.*0253

*And remember, the reason that we use order of operations is that, if we didn't, and everybody was just doing things their own way,*0259

*we couldn't really communicate using math, because people would write something down,*0265

*and somebody might do it in a different order and come up with a different answer.*0269

*So, this way, it is an agreed-upon set of rules that everyone follows.*0273

*Monomials: a monomial is a product of a number and 0 or more variables.*0281

*Again, refreshing your memory from Algebra I: examples of a monomial would be 5y, 6xy ^{2}, z, 5.*0288

*So, it says it is a product of a number and zero or more variables.*0304

*Here, there aren't any variables, so that actually is simply a constant; but it is still called a monomial, also.*0307

*Here, I have 5 times one variable; here I have multiple variables.*0314

*This is examples of...these are all monomials.*0321

*A constant is simply a number; so, it could be -3 or 6 or 14; those are constants.*0328

*Coefficients: a coefficient is the number in front of the variable.*0346

*Up here, I said I had 5y and 6xy ^{2}: this is a coefficient: 5 is a coefficient, and 6 is a coefficient.*0363

*When you see something like z, it does have a coefficient: it actually has a coefficient of 1.*0372

*However, by convention, we usually don't write the 1--we just write it as z, but it actually does have a coefficient of 1.*0378

*Next, degree--the degree of a monomial: the degree of a monomial is the sum of the degrees of all of the variables.*0386

*So, it is the sum of the degree of all of the variables.*0394

*For example, 3xy ^{2}z^{4}: if I want to find the degree, I am going to add the degree of each variable.*0408

*This is x (but that really means x to the 1--the 1 is unstated) plus y ^{2} (the degree is 2), plus z^{4} (the degree is 4).*0418

*Adding these up, the degree for this monomial is 7.*0429

*When we talk about powers: powers refer to a number or variable being multiplied by itself n times, where n is the power.*0435

*For example, if I say that I have 5 ^{2}, what I am really saying is 5 times 5.*0447

*So, 5 is being multiplied by itself twice, where n equals 2.*0455

*I could say I have y ^{4}: that equals y times y times y times y; and here, the power is 4.*0461

*OK, continuing on with more concepts: a polynomial is a monomial or a sum of monomials.*0477

*Recall the concepts of term, like terms, binomial, and trinomial.*0485

*A polynomial is simply an expression in which the terms are monomials.*0492

*And we say "sum," but this applies to subtraction, as well--a polynomial can certainly involve subtraction.*0496

*For example, 4x ^{2}+x or 2y^{2}+3y+4: these are both polynomials.*0504

*We also could say that 5z is a polynomial, but it is also a monomial; there is only one term, so it is a polynomial, but we usually just say it is a monomial.*0519

*OK, so looking at these other words: a binomial is a polynomial that contains two terms.*0532

*So, it is the sum of two monomials, whereas the trinomial is the sum of three monomials.*0544

*A monomial is simply a single monomial.*0552

*Recall that, as discussed, a term is a number or a letter (which is a variable) or both, separated by a sign.*0561

*Terms could be a number; they could be a variable; or they could be both.*0572

*3x-7+z: here, I have a number and a variable; here, I just have a number (I have a constant); here, I just have a variable.*0582

*And they are separated by signs--by a negative sign and a positive sign--so each one of these is a term.*0594

*The concept of like terms is very important, because like terms can be added or subtracted.*0602

*Like terms contain the same variables to the same powers.*0609

*For example, 1 and 6 are like terms; they don't contain any variables, so they are like terms.*0626

*3xy and 4xy are like terms; they both contain an x to the first power and a y to the first power.*0636

*2y ^{2} and 8y^{2} are also like terms: they both contain a y raised to the second power.*0645

*And so, these can be combined: they can be added and subtracted.*0653

*A formula is an equation involving several variables (2 or more), and it describes a relationship among the quantities represented by the variables.*0661

*And we have worked with formulas previously: and just to review, one formula that we talked about is the Pythagorean Theorem.*0670

*That is a ^{2}+b^{2}=c^{2}, where c is the length*0678

*of the hypotenuse of a right triangle, and a and b are the lengths of the two sides.*0683

*And this tells us the relationship among the three sides of the triangle.*0689

*And that is really what formulas are all about, and really what algebra is all about: describing relationships between various things.*0697

*And of course, during this course, we are going to be working with various formulas.*0706

*OK, in this example, we are asked to simplify or evaluate an algebraic expression.*0711

*5x ^{2}...and they are telling me x=3, y=-3; so I have some x terms and some y terms.*0718

*My first step is to substitute: so, everywhere I have an x, I am putting in a 3; everywhere I have a y, I am putting in a -3.*0724

*So, here I have 3xy, so here it is going to be 3 times 3 times -3.*0736

*Recall the order of operations: the first thing I am going to do is to get rid of the grouping symbols.*0740

*Take care of the parentheses; and looking, I do have parentheses.*0746

*In here, I have a negative and a negative; so I am simplifying that just to positive 3.*0755

*OK, continuing to simplify inside the parentheses: 3 plus 3 is 6.*0766

*I completed my first step in the order of operations.*0775

*The next thing to do is evaluate powers; and I do have some terms that are raised to various powers.*0778

*3 ^{2} is 9, minus 2 times 6^{3}; so, 6 times 6 is 36, times 6 is 216.*0785

*That took care of my powers; and the next thing is going to be to multiply and divide.*0801

*And when we do that, we always proceed from left to right.*0807

*2 times 216 is 432; OK, now I have: 3 times 3 is 9; 9 times -3 is -27.*0812

*I am going to rewrite this as 9 minus 432 minus 27.*0825

*Finally, add and subtract; and this is going from left to right.*0830

*9 minus 432 gives me -423, minus 27 (so now I have another bit of subtraction to do--that is -423-27) gives me -450.*0839

*So again, the first step was substituting in 3 and -3 for x and y.*0854

*The next step was to get rid of my grouping symbols; evaluate the powers;*0860

*multiply and divide, going from left to right (and I just had multiplication here);*0867

*and then, add and subtract, going from left to right, to get -450.*0873

*In this second example, again, we are asked to evaluate an algebraic expression; and here, we have three variables: a, b, and c.*0879

*So, carefully substituting in each of these, a=-1...so -1 ^{2}, minus 2, times b (b is 2), times c (c is 3), plus 3^{3}.*0886

*Here, I have c ^{2} in the denominator; so that gives me 3^{2}, minus 2, times a (which is -1), times b (which is 2).*0907

*Since there was a lot of substituting, it is a good idea to check your work.*0918

*a is -1 (that is -1 ^{2}), minus 2, times b, times c, plus c^{3};*0923

*all of that is divided by 3 ^{2} (so that is c^{2}) minus 2, times a, times b.*0934

*Everything looks good; now, the first thing I want to do is eliminate grouping symbols.*0941

*Recall that, in this type of a case, the fraction bar is functioning as a grouping symbol.*0945

*So, the whole numerator should be simplified, and the denominator should be simplified; and then I should divide one by the other.*0951

*Starting with the numerator: within the numerator, there are not any grouping symbols,*0958

*so I am going to go ahead and go to the next step, which is to take care of powers.*0964

*And -1 times -1 is 1; and then, I have 3 ^{3}; that is 3 times 3 (is 9), times 3 (is 27).*0971

*OK, I can do the same thing in the denominator; I can just do these both in parallel.*0991

*And so, I am going to evaluate the powers in the denominator.*0997

*3 times 3 is 9; and then, I don't have any more powers--OK.*1000

*So, I took care of that; my next step is going to be to multiply and divide.*1005

*OK, so I have, in the numerator, 1 minus 2 times 2 (is 4), and then 4 times 3 (is 12), plus 27.*1016

*So, that took care of that step; now, in the denominator, I have 9, and then I have minus 2, times -1.*1035

*So, 2 times -1 is going to give me -2; -2 times 2 is going to give me -4.*1046

*OK, so now, I have taken care of all of the multiplication and the division.*1063

*The next step is to add and subtract--once again, going from left to right.*1068

*So, starting up here, the next step is going to be 1 minus 12; 1 minus 12 is going to give me -11.*1073

*So, it is -11 plus 27; that is going to leave me with 16 in the numerator.*1086

*In the denominator, I have 9, minus -4; well, a negative and a negative gives me a positive,*1092

*so in the denominator, I actually have 9 plus 4, which gives me 13.*1099

*The result is 16 over 13.*1109

*Again, starting out by substituting values for a, b, and c...I have done that in this first step.*1112

*And then, I treat this fraction bar as a big grouping symbol, and then I take care of the numerator and the denominator separately.*1120

*You could have done them one at a time, or you can do steps at the same time.*1128

*So, first, evaluate the powers; I did that in the numerator; I did that in the denominator (I am treating them separately).*1132

*Multiplying and dividing: I did my multiplication here and in the denominator.*1138

*And finally, adding and subtracting to get 16/13.*1146

*Example 3: The formula for the area of a triangle is Area equals 1/2 bh.*1152

*So, this is actually that the area equals one-half the base times the height of the triangle.*1158

*Find the height if a is 32, and the base (b) is 8.*1164

*OK, here we are being asked to find the height, and we are given the other two variables.*1173

*So, let me rewrite the formula: area equals 1/2 base times height.*1179

*Now, I am going to substitute in what I was given.*1183

*I am given the area; I am given the base; and I need to find the height.*1185

*What I need to do is isolate h; so, first simplifying this: 32=...well, 1/2 of 8 is 4, so that gives me...4h.*1195

*Next, divide both sides of the equation by 4 (32/4 and 4h/4) in order to isolate that.*1206

*Well, 32 divided by 4 is 8; the 4's cancel out on the right; and then just rewriting this in a more standard form, with the variable on left, the height is 8.*1216

*So again, first just write out the formula; substitute in a and b (which I was given).*1228

*And then, solve for the height.*1237

*The temperature in Fahrenheit is F=9/5C+32, where C is the temperature in Celsius.*1242

*If the temperature is 78 degrees Fahrenheit, what is it in Celsius?*1250

*Rewrite the formula and substitute in what is given.*1255

*What is given is that the temperature in Fahrenheit is 78.*1263

*And I am looking for Celsius (I always keep in mind what I am looking for--what is my goal?).*1267

*And that is +32; my goal here is going to be to solve for C--to isolate that.*1272

*Subtracting 32 from both sides gives me 46=9/5C.*1281

*Now, in order to isolate the Celsius, I am going to multiply both sides by 5/9.*1290

*When I do that, I am going to get 5 times 46 (is 230), and that is divided by 9.*1302

*Here, that all cancels out; so, rewriting this, Celsius equals 230/9.*1310

*That is not usually how we talk about temperature; so simplifying that, if I took 230 and divided it by 9, I would get approximately 25.5 degrees Celsius.*1317

*So again, the formula for converting Fahrenheit into Celsius (or vice versa) is given.*1327

*I substituted in 78 degrees and figured this out: so, 78 degrees would be equal to approximately 25.5 degrees Celsius.*1335

*That concludes today's lesson on Educator.com; and I will see you again for the next lesson.*1346

1 answer

Last reply by: Dr Carleen Eaton

Wed Apr 19, 2017 5:54 PM

Post by Anna Kopituk on April 17, 2017

Hi Dr. Eaton,

Thank you for making Algebra understandable. I am a homeschool student and my Mom is wondering how to test me on this information? Do you have tests available or know how we can find tests that fit your syllabus/curriculum?

Thank you,

Wesley

0 answers

Post by Khanh Nguyen on October 19, 2015

Practice Question 8 is wrong, the answer should be -3, not 3.

0 answers

Post by julius mogyorossy on July 14, 2013

You should say terms are anything separated by +, - or =, the difference between terms and factors really confused me at first, the different rules.

2 answers

Last reply by: Lexlyn Alexander

Thu Oct 23, 2014 5:37 PM

Post by Manika Marwah on May 30, 2013

does n denote power eg power of 3 is 6 so n=6??

0 answers

Post by Aniket Dhawan on May 5, 2013

hi

2 answers

Last reply by: Jeremy Canaday

Thu Aug 8, 2013 1:16 PM

Post by Joyce Andrews-McKinney on January 24, 2013

I like E.com fairly well. I use this site as a guide to teach math to my home schooler. I am confused as to why this site teaches Algebra 11 BEFORE teaching Geometry when all other schools and programs teach Geometry then Algebra 11. Does it matter which follows Algebra 1? I really need a response. Thanks everyone.

3 answers

Last reply by: Jonathan Aguero

Fri Dec 7, 2012 3:00 PM

Post by Sheila Mckenzie on April 29, 2012

When you were converting degrees to celcius. Why did you inverse the fraction on both sides from 9 over 5 to 5 over 9. Is it because its a fraction and the rule is when your multiplying a fraction on both sides you inverse the numbers? Please advise.

3 answers

Last reply by: Yih S.

Fri Aug 5, 2016 2:12 PM

Post by javier mancha on August 23, 2011

do the exponents, get added as a degree ??