Advanced Algebra Semester 2 Final Review

Guide to the Semester 2 Final Concept #1: Matrices

(1/31/07)

Reduced Row Echelon Form:

Row Operations: #1: Multiply or Divide all #’s in a row by a non-zero # #2: Add two rows together #3: Add a Multiple of the #’s in one row to the corresponding #’s in another row. #4: Exchange rows -2.5 •R1 + R2 2 •R2 -R2 + R1 .5 •R1

R2 R1 R1

Matrix Operations:  To add or subtract matrices they must have the exact same # of rows and columns  To multiply matrices the internal dimensions must be the same

Ex.

You can multiply a [2•2] • [2•3] You can’t multiply a [2•3] • [2•3]

Addition Example:

Multiplication Example:

•

=

Identifying the Identity:

Example:

-

2A+C=1 4A + 3C =0 C= (-2)

-

A= (3/2)

2B+D=0

=

4B + 3D =1 B= (-1/2)

D= (1)

Identity Matrix:

Matrix •[A]-1 = [I] Solving Systems with Inverse Matrices = 4A + 2C =1 2A – 3C =0 A=3/16, B=1/8

= 4B + 2D = 0 2B-3D = 1 B=1/8, D=/1/4

= And guess what???? You don’t need to know how to solve matrices with three rows, cause of CRAMER’S RULE !! so without further ado, Cramer’s Rule: Cramer’s Rule Second order determinants first: a square array of #s evaluated according to the following rule: ax +by=c dx +ey =f

Third order determinants “Expansion by Minors”

=

– NR

by using:

where

Conic Sections: Distance Formula:

Equation of a Circle: R2 = (x-h) 2 + (y-k) 2 Equation of an Ellipse:

To find the focal points use the following equation: a 2 - b 2 = c2 Where “a” is the larger radius and “b” is the smaller radius, and “c” is the distance from the center along the major axis to the focal pt. Equation of a Hyperbola:

To find the focal points use the equation: a 2 + b 2 = c2 Where “a” is the larger radius and “b” is the smaller radius, and “c” is the distance from the center along the major axis to the focal pt

To find asymtotes: Use the ratio 1. Hyperbolas always open in the direction of the positive squared term

Parabolas: X= ay2 + by +c formula: -b/2a

** to find the y coordinate of the vertex use the

Discriminants:

(when there is an xy term in

standard form) D<0

= Ellipse, Circle

D=0

=Parallel Lines, Parabola

D>0

= Hyperbola

Rational Algebraic Functions: General form: *Where p(x) and q(x) are polynomials Two types of discontinuities: 1. 0/0 is a removable discontinuity or hole 2. x/0 where x is not a zero is an asymtote

Rational Root Theorem: If the polynomial equation p(x) = 0 has rational roots they are of the form p/q where p is a factor of the constant term and q is a factor of the leading coefficient. ****** remember to give coordinates for holes/ removable discontinuities ******remember to simplify all factors Rational Root Theorem: If the polynomial equation p(x) = 0 has rational roots they are of the form p/q where p is a factor of the constant term and q is a factor of the leading coefficient. Use polynomial or synthetic division

Study Sheet: Chapter 11 ’Series’

11.1 Arithmetic Series:

6/2/08

Notation: The Greek letter sigma represents summation notation

For arithmetic sequences the equation to calculate partial sums is

Sn = n/2 (t1 + tn) Where t1 is the first term and tn is the last term Ex.

To solve, find the number of values you are calculating the partial sum for 20 Calculate the value of term 1 = 15 Calculate the value of term 20 = 205 Plug it into the formula, and you get 20/2(15+205) when you solve, you get 2200.

11-2 Infinite Geometric Sequences 6/4/08 Convergent geometric series: same idea as a shifted geometric sequence (see semester 1) An infinite geometric series in convergent if the absolute value of the common ratio is less than 1. In a geometric series given |r|< 1, the value that the series converges to is found with the formula:

11-3 Partial Sums of Geometric Sequences 6/6/08 To find the partial sum of a geometric finite series use the formula:

*Where t1 is the first term, r is the ratio, and n is the number of terms *****S refers to sum, U refers to the term*****

Simple Radical Form •

No radicals in the denominator

•

Root index is as low as possible

•

Radicand of the nth root has no nth powers as factors Ex.

Remember holes and asymtotes: we didn’t have time to explain all of it Holes: 0/0

Asymtotes: x/0 where x isn’t 0 Solving Fracitonal Equations 1. Write the domain : what #’s aren’t included 2. Multiply all terms by the smallest expression needed to eliminate the fractions 3. Solve the polynomial 4. Discard extraneous solutions 5. Write the solution set

Descartes Rule of Signs: The # of positive roots of a polynomial is less than or equal to the # of positive sign changes of f(x) The # of negative roots of a polynomial is less than or equal to the # of negative sign changes of f(-x) Non-real roots must come in pairs of two.

REMEMBER TRIG!!! Soh cah toa Sin= oppositive/hypotenuse Cosine= adjacent/hypotenuse Tangent= opposite/adjacent