INSTRUCTORS Carleen Eaton Grant Fraser

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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

Evalaute the expression x3 − 2x2 − 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
− 5
Evaluate the expression [(x2 + 3x − 3)/(y2 − 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)/(32 − 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
− [5/7]
Classify the polynomial 3x2 + 3x + 3 as monomial, binomial or trinomial and state the power of each term.
• The polynomial 3x2 + 3x + 3 is a trinomial because it has 3 terms.
• The power of the first term = 2
• The power of the second term = 1
The power of the third term = 0. The power is zero because recall that any number or variable raised to the zero power is always 1, therefore 3 can be written as 3x0 = 3.
Classify the polynomial − [3/5]x2y3z5 as monomial, binomial or trinomial and state the power of each term.
• The polynomial − [3/5]x2y3z5 is a monomial because it only has one term.
In this case, the power is determined by the highest power in the term, therefore, the power = 5.
Classify the polynomial x2 − 25 as monomial, binomial or trinomial and state the power of each term.
• The polynomial x2 − 25 is a binomial because there are 2 terms.
• The power of the first term = 2.
The power of the second term = 0. It's zero because any variable or number raised to the zero power is always 1.
A ball is thrown upwards into the air at a speed of 64ft/s. The formula h = − 16t2 + 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
After two seconds, the ball is 64 ft from the ground.
Simplify the expressiong by following the rules of operations. [3 + 2(5 − 2)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
400
Simplify the expression using the rules of exponents. 3(2 + (3 − (5 − 3)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.
3(−1) = −3
Evaluate the polynomial x5 + x3 + 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
A negative base raised to an odd power is always negative.
Evaluate the polynomial x4 + x2 + 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
A negative base raised to an even power is always positive.

*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.

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 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
• 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

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 y2.0065

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, 4x3+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 42 or 34), 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, 6xy2, 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 6xy2: 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, 3xy2z4: 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 y2 (the degree is 2), plus z4 (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 52, 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 y4: 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, 4x2+x or 2y2+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

2y2 and 8y2 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 a2+b2=c2, where c is the length0678

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

5x2...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

32 is 9, minus 2 times 63; 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 -12, minus 2, times b (b is 2), times c (c is 3), plus 33.0886

Here, I have c2 in the denominator; so that gives me 32, 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 -12), minus 2, times b, times c, plus c3;0923

all of that is divided by 32 (so that is c2) 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 33; 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