Math Analysis - Loudoun County Public Schools



Math Analysis Name: ____________________________

Review Packet Notes I

Solving Equations and Inequalities

You must have a firm grasp on how to solve all types of equations. I expect that you already know how to solve one, two, and multi-step equations and will not provide notes on them here. Also, be sure you show each step of your work. NO WORK, NO CREDIT!

Inequalities

Don’t forget about working with inequalities. You need to remember to flip the inequality symbol when dividing by a negative.

EX] 2x ( -18 EX] -2x ( -18

x ( -9 DON’T FLIP x ( 9 FLIP

Absolute Value Equations

EX] (2x + 3( = 5

2x + 3 = 5 AND 2x + 3 = -5 Remember to set up 2 equations!

2x = 2 AND 2x = -8 Subtract 3 from both sides

x = 1 AND x = - 4 Divide each side by 2

Radical Equations

EX] [pic] Original Problem

[pic] Isolate radical by adding 5 to both sides

[pic] Cube each side

8x + 3 = 27 Simplify

8x = 24 Subtract 3 from both sides

x = 3 Divide both sides by 8

NOTE: Be sure that you check your solution(s)! This is one of the best ways

to catch yourself making those silly little careless mistakes that drive you

and your math teachers crazy! In the case of radical equations, your

solutions may be extraneous.

SYSTEMS OF EQUATIONS

You should know how to solve a system of equations in two-variables by graphing, substitution, linear combination (also called elimination), and matrices. Systems involving more than two variables are done using the calculator.

NOTE: When solving systems, you can get one of the following answers:

• 1 solution (written as a coordinate point)

• infinitely many solutions

• no solution.

SOLVING QUADRATIC EQUATIONS

This single concept is the bulk of Algebra II. The terms roots, zeros, x-intercepts, and solutions are all synonymous. You should know how to solve quadratics by factoring, completing the square, and of course, the quadratic formula (please, please, please, memorize this!)

The quadratic formula: [pic]

EX] y = x2 – x – 6

y = (x + 2)(x – 3) Factor

x + 2 = 0 x – 3 = 0 Set each factor equal to 0

x = -2 x = 3 Solve

EX] y = x2 – x – 6 Let a = 1, b = -1, and c = -6

[pic] Substitute into quadratic formula

[pic] = [pic] = [pic] Simplify

[pic] AND [pic] Simplify

DON’T FORGET TO CHECK YOUR ANSWERS!

POLYNOMIAL FUNCTIONS

The Rational Zero Theorem

When solving polynomial equations, it is important to know the rational zero theorem:

If f(x) = anxn + … + a1x + a0 has integer coefficients , then every rational zero has the form

[pic]

EX] Find the rational zeros of f(x) = 3x3 – 4x2 – 17x + 6

Possible rational zeros: [pic]

Simplifying yields the following possibilities: ( 1, ( 2, ( 3, ( 6, [pic]

We then use either the calculator (look in the TABLE) or synthetic division to test the possible zeros. You may have to test several before you find one that works. Here, x = -2 works:

-2 3 -4 -17 6

-6 20 -6

Remember how to get the

3 -10 3 0 factor of f(x) from here?

Since x = -2 is a zero, then x + 2 is a factor of f.

f(x) = (x + 2)(3x2 – 10x + 3) Factor original polynomial

f(x) = (x + 2)(3x – 1)(x – 3) Factoring the trinomial

Setting each factor equal to 0 and solving gives us the zeros.

x + 2 = 0 3x – 1 = 0 x – 3 = 0

x = -2 3x = 1 x = 3

The zeros are x = -2, x = [pic], and x = 3

Factor by Grouping

Another skill to use when finding rational zeros of a polynomial function is factor by grouping.

f(x) = 2x3 + 2x2 – 8x – 8 Original equation

f(x) = 2x2(x + 1) – 8(x + 1) Factor GCF’s

f(x) = (2x2 – 8)(x + 1) Rewrite

f(x) =2(x2 – 4)(x + 1) Factor

The zeros are –2, 2, and –1.

NOTE: Be careful here! A common mistake is to forget the (

x2 – 4 = 0

x2 = 4

x = ( 2

Writing Polynomial Equations from Zeros

Given the zeros of a polynomial function, you should be able to write the equation of the original polynomial equation. The main thing here is to watch your signs!

EX] The zeros of a polynomial function are –1, 5, and 6. Write the function of least degree.

f(x) = (x + 1)(x – 5)(x – 6)

f(x) = (x2 – 4x – 5) (x – 6) Use FOIL with first two terms

f(x) = x3 – 4x2 – 5x – 6x2 + 24x + 30 Multiply

f(x) = x3 – 10x2 + 19x + 30 Combine like terms

NOTE: Use the TI-83 to check

Put original in Y1 and your answer in Y2 Look at your table of values –

They should be the same!

RATIONAL EXPRESSIONS

I expect that you already know how to simplify rational expressions, as well as multiply and divide them. You need to know how to factor!

Adding and subtracting rational expressions is just like adding and subtracting fractions – you must have the same denominator!

EX] Add: [pic] The LCD is 6x2

= [pic] Rewrite the fractions with LCD

= [pic] Simplify and add numerators

Simplifying Complex Fractions

Simplifying complex fractions just takes patience. You have to watch your algebra as you go through these kinds of problems.

EX] Simplify [pic]

[pic] = [pic] Rewrite fractions in numerator with LCD (1)

= [pic] Distribute (2)

= [pic] Simplify numerator (3)

= [pic] Factor numerator (4)

= [pic] Multiply by reciprocal (5)

= [pic] Write in simplified form (6)

NOTES: 1) There is no need to distribute in line (6) above

2) There is more than one way to do this problem. I like to simplify

the numerator as much as possible before I begin to work with

the denominator.

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