Mrs. Nelson’s Little Black BookTable of Contents Lin ear Equa tio ns …………………………………………………………………………………………. 2 Abso lute Value Equa tion s ………………………………………………………………………………... 4 Piecewise Func tions ………………………………………………………………………………………. 6 Simpl ifyi ng Poly nomia ls …………………………………………………………………………………. 8 Scien tific Nota tion ………………………………………………………………………………………… 9 Fac tor ing ………………………………………………………………………………………………….. 11 Ratio nal Expr essio ns ……………………………………………………………………………………… 12 Synthetic Division ………………………………………………………………………………………… 14 Radic al Expr essio ns ………………………………………………………………………………………. 17 Compl ex Numbe rs ………………………………………………………………………………………… 20 Deriving the Quadratic Formula …………………………………………………………………………… 22 Exponential/L ogarithmic Equations ……………………………………………………………………….. 23 Syste ms of Equati ons -2va riabl es …………………………………………………………………………. 27 Systems of Equati ons -3va riabl es …………………………………………………………………………. 29 Matri ces …………………………………………………………………………………………………… 32 Bin omi al The orem ………………………………………………………………………………………. 36 Cou nti ng and Probab iltiy ………………………………………………………………………………… 38 Sig Fig s …………………………………………………………………………………………………… 41
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Absolute Value Equations
Definitions:
Absolute Value: the distance of “x” from zero
Absolute Value Equation: an equation that involves the absolute value of a variable expressionAbsolute Value Inequalities: an inequality that involves the absolute value of a variable expression
Examples:
Ex 1: Solve 712 =+ x
-1st step with any absolute value equation is to clear the absolute value bars1. 2x+1 = 7 or 2x+1= -7 -answer could be either 7 or -7 since variable
was
inside the absolute value bars
2. 2x= 6 or 2x= -8 -finish solving like a regular equation
3. x=3 or x= -4
4. {-4, 3}
Ex2: Solve 712 >+ x
1. 2x+1> 7or 2x+1< -7
- re-written because 2x+1 bust represent a number more than 7 units from 0 on
the
number line.
2. 2x> 6 or 2x< -8
3. x>3 or x< -4
4. (-∞, -4) U (3, ∞)
Ex 3: Solve 712 <+ x
1. -7<2x+1<7
-re-written because it must be less than 7 units from 0, so must be between -7
Rules for Multiplication in Scientific Notation:1) Multiply the coefficients2) Add the exponents (base 10 remains) (Multiplication Using Scientific Notation)
3) Adjust the final power of 10 if the decimal needs to be re-placed behind the first non-zero digit.
Rules for Division in Scientific Notation:1) Divide the coefficients2) Subtract the exponents (base 10 remains) (Division Using Scientific Notation)
3) Adjust the final power of 10 if the decimal needs to be re-placed behind the first non-zero digit.
Problem A: convert to scientific notation
4760Step 1: place asterik to right of 4
Step 2: count spaces to decimal point
Step 3: exponent is positive because asterik is
to left of original decimal point
3
1076.4 ×Step 1: 760*4
Step 2: 4*760
3 spaces
Step 3: 31076.44760 ×=
Problem B: convert to scientific notation
.00091
Step 1: place asterik to right of 9
Step 2: count spaces to decimal point
Step 3: exponent is negative because asterik is
to right of original decimal point
4101.9 −
×
Step 1: .0009*1
Step 2: .0009*1
4 spaces
Step 3: .00091 = 4101.9
−×
Problem C: convert from scientific notation31,000,900
Step 1: Move decimal 7 points to the right of itscurrent position
References"Multiplication using Scientific Notation." Edinformatics -- Education for the Information Age. 04
May 2009 <http://www.edinformatics.com/math_science/scinot_mult_div.htm>.
“Scientific Notation Using Division." Edinformatics -- Education for the Information Age. 04 May2009 <http://www.edinformatics.com/math_science/scinot_mult_divb.htm>.
a. Identify the following parts of function notation:
• (x-k) = divisor• g(q) = quotient (answer)• r = remainder **always +, if the remainder is negative enclose it in ( )
Sometimes you will be asked for the answer only, not in function notation: r q g +)(
• g(q) = quotient (answer)• r = remainder (written over the divisor)
• This is sometimes called mixed number format.b. Synthetic Division: a shorthand, or shortcut, method of polynomial division in
the special case of dividing by a linear factor (and it only works in this case).Synthetic division is generally used, however, not for dividing out factors but forfinding zeroes (or roots) of polynomials. (Staple, 1998)
c. To find zeroes of polynomial equations: If you are given the polynomialequation y = x 2 + 5x + 6, you can factor the polynomial as y = ( x + 3)( x + 2).
Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x = –2 and x = –3 are the two zeroes of y . (Staple, 1998)d. ***It is always true that, when you use synthetic division, your answer will be
raised to a power one less than what you'd started with.e. Step-by-Step Instructions:
Written Steps: Math Steps:
1. Original Problem:
2. First, carry down the “2” that indicates the
leading coefficient.
Write k on the outside, and the coefficientson the inside.
**Remember that in the form ( x-k ), that the –
is part of the form so adjust the signaccordingly.
Radical Expressions By Mrs. Lenora Nelson 1/27/09 From www.freewebs.com/mrs--nelson
#1: Definitions & Rulesa. Identify the following parts of the radical expression 3 27
Index = 3
Radical =
Radicand = 27 b. Define:
Principal root: The positive root of a square root. If asked to find “the” root, principal
root is implied. If asked to find “all” roots then you would want the positive andnegative roots.
Following are true statements regarding radical expressions.
(CliffNotes, 2009)
c. Rules for Radicals
Simplifying : To simplify numbers that aren’t perfect squares you must factor into
perfect squares and work from there.
Adding/Subtracting: The index and radicand must be the same. If you have numbers in
front of the radical, treat as coefficients and obey integer rules. The radicand acts as the
variable and remains unchanged.
Multiplying/Dividing: The index must be the same. If you have numbers in front of theradical, treat as coefficients and mult/div obeying integer rules. The radicands are also
mult/div within the radical and then simplified if possible.
Radicals in Denominator: These are not allowed. You must rationalize thedenominator by multiplying by something equivalent to 1. See example
those without the i. Deal with any parentheses and
combine like terms.
Ex. 2Here is an example with a subtraction part. Notice
the negative was distributed over the parentheses.
Ex. 1
i
ii
ii
78
4632
)46()32(
+
+++
+++
Ex. 2
i
i
i
34
395
)39(5
+−
+−
−−
Multiplying Complex Numbers
Ex. 1
When you multiply complex numbers, you have
to factor them. Remember FOIL, so that way youget all the right numbers. Once you've FOIL'd,
you can combine like terms. Remember from the
definitions earlier, that i^2 is equal to a -1 and
since -1 * -10=10, you add 10 instead of subtract10.
Ex. 1
i
i
iii
ii
1422
101412
1020612
)24)(53(
2
+
++
−+−
−+
Dividing Complex Numbers
Ex. 1When you divide, you have to remember the
conjugate, which is defined above. You multiply
both parts of the problem by the conjugate andthen foil. You put those answers over each other,
and then divide like normal.
Ex. 1
i
ii
i
i
i
i
+
+=
+−−
++
2
29
)2(29
29
2958
25
25*
25
98
Calculator HelpMany calculations with i can be performed using the i button on your TI-84 calculator. The i button is locatedon the decimal key. Don’t forget to use parentheses.
Simplifying Powers of i by Hannah 08-09 Pattern even exponents of i :
• If the exponent divided by 2 equals
an even number, then it is +1
• If the exponent divided by 2 equals
an odd number, then it is -1i2= -1 i4=1
i6= -1 i8=1
i10= -1 i12=1
i14= -1 i16=1
i18= -1 i20=1
Pattern odd exponents of i:
• If the exponent divided by 2 equals
an even number, then it is +i
• If the exponent divided by 2 equalsan odd number, then it is – i
Exponential and Logarithmic Functions By Mrs. Lenora Nelson 3/2/09 From www.freewebs.com/mrs--nelson
#1: Definitions & Rulesd. Identify the following parts of this logarithm 125log5= y
Exponent = y
Yielded Number = 125
Base = 5e. Define:
One-to-One Function: a function in which each x-value corresponds to only one y-value and
vice versa (Lial & Hornsby, 2000, p. 634). None of the x’s or y’s repeat.
Exponential Function: a function with a variable in the exponent. F( x) = a x where a>0 and
1
Logarithm: the inverse of an exponential function. It describes the exponent needed to
produce a given answer. x y alog= (a ≠ 1, a>0, x>0 ). 125log5= y With the base of 5 a 3
needed to yield a 125.
The red line shows the exponential function.The green line is the line of reflection.The blue line shows the logarithmic function.
Common Logarithm: Logarithms to the base10. Calculators evaluate Logs base 10.
Natural Logarithms: “The logarithm base e of a number. That is, the power of e necessary t
equal a given number. The natural logarithm of x is written ln x. For example, ln 8 is2.0794415... since e2.0794415... = 8” (Simmons, 2006). They are called natural because they occ
in biology and the social sciences in natural situations that involve growth or decay (Lial &
Hornsby, 2000, p. 672). All of the Logarithm Rules below apply to ln as well.
e : e ≈ 2.7182818284.... is a transcendental number commonly encountered when workingwith exponential models of growth, decay,and logistic models, and continuously compound
interest (Simmons, 2006).
e is the unique number with the property that the area of the region boundethe hyperbola , the x-axis, and the vertical lines and is 1.
other words,
With the possible exception of , is the most important constant in
mathematics since it appears in myriad mathematical contexts involving lim
How to determine if One-to-One Function: If the graph of the function passes the horizonta
line test (no 2 points touch the function) then it is one-to-one. If you are given an equation
format you recognize apply the horizontal line test otherwise graph first.
How to find the inverse of a one-to-one function:
1. First verify that it is one-
to-one.2. Swap the x and the y.
3. Solve for y
4. Write in the format
...)(1 =− x f
3 1)( += x x f
3 1+= y x
1
13
3
−=
+=
x y
y x
1)(1 +=− y x f
Product Rule: baab logloglog +=
Quotient Rule:
bab
alogloglog −=
Power Rule: aa log2log
2=
Change of Base Rule:2log
9log8log 2 = This example shows taking a log base 2 and converti
to log base 10. This is most commonly done to find a numerical value using a calculator orgraph logs other than base 10 in a calculator. The Change of Base Rule can be used to conv
to any base if needed.
#2: Sample Problems: Solution DescriptionsProblem a: Rewrite in exponential form
Step 1: Write the base of 49 raised to the ½. Place the
yield number on the other side of the = sign.
Original problem:2
17log49 =
Step 1: 749 2/1 =
Problem b: Rewrite in exponential form
Step 1: ln is the natural logarithm to the base of e so raise
t to the 2nd. Place the yield number x on the other side of
he = sign.
Original problem: 2ln = x
Step 1: xe =2
Problem c: Rewrite in logarithmic form.
Step 1: Write the base as a log to base 81. Place the yieldnumber next. Then place the exponent number ½ on the
other side of the = sign.
Original problem: 981 2/1 =
Step 1: 219lo g81 =
Problem d: Rewrite in logarithmic form.
Step 1: Write the base e as ln. Place the yield number
next. Then place the exponent x on the other side of the =ign.
Simmons, B. (2006). Definition of e. Mathwords: Terms and Formulas from Algebra I to Calculus. Retrieved Marc
6, 2009, from http://www.mathwords.com/e/e.htm.
Simmons, B. (2006). Natural Logarithm. Mathwords: Terms and Formulas from Algebra I to Calculus. RetrievedMarch 6, 2009, from http://www.mathwords.com/n/natural_logarithm.htm.
Sondow, J. & Weisstein, E. W. (n.d.). “e.” MathWorld --A Wolfram Web Resource. Retrieved March 6, 2009, from
Systems of Equations: 3 Variables (Ch. 11.2)By: Hannah 08-09
A. Definitions:
1. ordered triple- a solution of an equation in three variables, such as 2x + 3y –z = 4, is called an ordered tripl
and is written (x,y,z)
2. system of equations- a set of equations in which finding the numbers makes 2 or more equations true at the
same time3. linear system- 2 or more linear equations form a linear system
4. solution set of a linear system- contains all ordered pairs that satisfy all the equations of the system at the s
time
Lial & Hornsby, 2000)
B. The graph of a linear equation with 3 variables is a plane not a line.
Graphs of Linear Systems in Three Variables
1. The 3 planes may meet at a single, common point that is the solution of
the system.
2. The 3 planes may have the points of a line in common so that the set of
points that satisfy the equation of the line is the solution of the system.
3. The planes may have no points common to all 3 so that there is nosolution for the system.
4. The 3 planes may coincide so that the solution of the system is the set of all points on a plane.
Lial & Hornsby, 2000)C. Steps
Solving Linear Systems in 3 Variables by Elimination
Step 1: Eliminate a variable. Use the elimination method to eliminate any variable
from any 2 of the given equations. The result is an equation in 2 variables.
Step 2: Eliminate the same variable again. Eliminate the same variablefrom any other two equations. The result is an equation in the same twovariables as in Step 1.
Step 3: Eliminate a different variable and solve. Use the elimination method to
eliminate a second variable from the two equations in two variables that result from
Steps 1 and 2. The result is an equation in one variable that gives the value of thatvariable.
Step 4: Find a second value. Substitute the value of the variable found in Step 3
into either of the equations in two variables to find the value of the second variable.
Step 5: Find a third value. Use the values of the two variables from Steps 3 and 4
to find the value of the third variable by substituting into any of the originalequations.
Step 6: Find the solution set. Check the solution in all of the original equations.Then write the solution set.
• Multiplication with two matrices : The first matrix’s number of columns must match the second matrix’s numof rows. The numbers that are left will tell you the size of the answer.
Example:
−
•
−
23
32
46
405
243
• The first matrix is 2x3
• The second matrix is 3x2
o 2 x 3 3 x 2
Because the numbers on the inside (the threes) match, this can be multiplied. Because the numbers on the outside are both twos, the answer will be a 2x2 matrix
Once you have determined that the matrices can be multiplied, you multiply each element in the first row of first matrix by the elements in the first column of the second matrix. You add these answers together to get p
You repeat this procedure with the first row of the first matrix and the second column of the second matrix.
You repeat this procedure again with the second row of the first matrix and the first column of the secondmatrix, then once more with the second row of the first matrix and the second column of the second matrix.
matrices, wherein the answer of themultiplication problem would be the
original A.
I=
100
010
001
2. DETERMINANTS
86
43−
= A
-24 – 24 =
-48
2.
We start off by multiplying the top left
numeral with the bottom left numeral andget a product of negative 24.
Then, we multiply the bottom left numeralwith the top right numeral and get a
product of 24.
After, we subtract the second product from
the first and find that the determinant of
this matrix is -48.
While reading these steps, you may be asking yourself “But when will I ever need this inreal life?” Well, believe it or not, but many career path involve matrices. In small
businesses, matrices are used to compute revenues. Because of the fact that, in a store,
different products have different prices, matrices are great for figuring out costs, revenues,and profits of many different sales. In the same sector, matrices are also used to determine
production costs for similar products. Matrices are also used in cryptography as well as
What if the letters in the parenthesis already have coefficients or there aren’t letters, but numbers?
t’s simple really. When you make your spaces with the Pascal’s Triangle coefficients already attached to them, you
are ready to make the next step. Now do exactly as you did before but place anything with a coefficient (watch out fohe sign, if it’s a negative include that too) inside of parenthesis next to its correct Pascal Coefficient. Leave the
xponent at the end of the parenthesis. Once you’ve expanded all of that with both parts of the binomial then you mdistribute your exponents, coefficients and signs. This will give you the Binomial expansion.
Counting & ProbabilityBy: Jacqueline -- May 4th, 2009
Definitions:
• Fundamental principle of counting: If one event can occur in m ways and a second ev
can occur in n ways, then both events can occur in mn ways, provided the outcomethe first even does not influence the outcome of the second. (Hornsby, 2000)
1. First determine that the ordermatters, therefore this is apermutation.2. Because r is 7, you must
multiply 9 by its descendingnumbers 7 digits.3. Solve.
Problem 1: How many ways can 7people be lined up in a row from aclass of 9 people?
1. 9P7
2. 9(8)(7)(6)(5)(4)(3)
3. 181,440Problem 2: 15
1. Because the students will be ona committee with no assignedpositions, this is a combination.2. Set up the problem. Because r
is 4, you must multiply 6 by itsdescending numbers 4 digits.3. Cancel any repeating numbers.4. Solve.
Problem 2: Six students want tobe on the Senior Prom Committee.If only four students are permittedon the committee, how manydifferent ways can they bechosen?
1.
4
6
2.)1)(2)(3(4
)3)(4)(5(6
3.)1)(2(
)5(6
4. 15
Problem 3: 26
1
1. Because there are only 2 blackkings in a standard deck of cards(52), place 2 over 52.
2. Reduce for answer.
Problem 3: A card is drawn from astandard deck of 52 cards. Findthe probability of a black king.
1.52
2
2. 26
1
Problem 4:13
4
1. Find the probability of drawinga jack or a spade. Because thereare 4 jacks in a deck, place 4 over52. Because there are 13 spadesin a deck, place 13 over 52.2. Find the probability of arepeating card: a jack of spades.Because there’s only one jackthat’s also a spade, place 1 over52.
3. Add the probabilities of the
Problem 4: A card is drawn from astandard deck of 52 cards. Findthe probability of drawing a jack ora spade. (Hornsby, 2000)
Significant Figures are the minimum amount of digits required to report a valuewithout loss of accuracy. “It is important to use significant figures when recording a
measurement so that it does not appear to be more accurate than the equipment is capaof determining.”(Fetterman, 2007) For example, when using a ruler that only measuresinches (it doesn’t have little marks for tenths of inches) you can’t measure somethingaccurately to the thousandth’s place because the ruler doesn’t go up that high.
Significant Digits
Rule for DeterminingSig Figs
Example# of SigFigs
Nonzero digits are always significant 257 33.5 2
Digits to the left of a decimal point arealways significant
92. 2
360. 3
Zeroes between significant figures aresignificant
1,003 4
6.0001 5
Placeholders (0’s that indicate positionof a decimal point) are NOT significant
210 2
0.003 1
Zeroes at the end of a decimal pointare significant
0.320 3
0.0045000 5(Louisiana iLEAP)
Problem Solution1.) How many significant figures are in the number
34.0900?
1.) Using the rules for determining sig figs, the answer to
the problem is 6. We know this because all nonzero digits
are sig figs, and 0’s between sig figs are significant, and
also numbers that come after the decimal after a sig fig aresignificant as well, so all of the numbers in the problem are
significant.
2.) How many significant figures are in the number
0.0000000030?
2.) The answer to this problem is 2 because when you use
the rules for determining whether figures are significant or
not, only 2 follow the criteria. Because there are no sig figs
in front of the decimal point, all of the zeroes in front of the
three are NOT significant because they don’t have sig figs
on both sides of them. And as for the sig fig at the end, it is