Regents Review Session #1 Polynomials Adding and Subtracting Polynomials What do you do? 1. Add/subtract like terms Example: 1. (8x3 - 9x2 + 6x + 2) - (-7x3 - 5x2 + 1x - 8)

Multiplying With Polynomials What do you do? 1. Distribute (or double-distribute/FOIL, when necessary) 2. Combine like terms

Examples: 1. 4x2 - 6x(2x - 7) + 5 - 8x(x + 2)

3. (7x - 2)2

2. (4y - 3)(5y2 + 3y - 10)

**** 4.

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Factoring What do you do? 1. Always look for a greatest common factor first. 2. Factor by into two binomials - be sure those binomials are also in simplest factored form. (Remember - if there is no greatest common factor in the given trinomial, there cannot be a common factor in the parentheses) ­ Be aware of the difference of perfect squares, sum/difference of two cubes, and factoring by grouping. Examples: 1. x4 - 16

2. 6x3 + 14x2 - 9x - 21

3. 4x2 - 5x - 6

4. 75x3 - 200x2 - 3x + 8

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Simplifying Algebraic Fractions What do you do? 1. Factor the numerator, factor the denominator, and cancel common factors between the numerator and denominator. Examples: b. 6x2 - 5x - 4 10-2x a. 9x2 - 16 x2 - 2x - 15

c.

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Multiplying and Dividing Algebraic Fractions What do you do? 1. Factor EVERYTHING 2. Cancel any common factors that appear in a numerator and a denominator. REMEMBER: You are only allowed to cancel factors, factors are connected by a multiplication sign. YOU CANNOT CANCEL ACROSS AN ADDITION OR SUBTRACTION SIGN. 3. Multiply numerators, leave in factored form, and then multiply denominators.

** IF you are dividing, you must turn the division into multiplication and take the reciprocal of the 2nd fraction.

Examples:

4

2.

Express in simplest form:

3. Multiply x3 + 4x2 - 9x - 36 * x2 - 2x - 15

2x - 10 3x - x2

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Adding and Subtracting Algebraic Fractions What do you do? Factor all denominators. 1. 2. Find the LCD between all of the denominators. 3. Make the LCD the denominator on all fractions and appropriately adjust the numerators. 4. Keep the common denominator and add/subtract the numerators. 5. Simplify the expression, if possible. Examples: + x 1. x-3

2 x

2.

5x + 7 x2 - 4

-

3 4x + 8

6

3. 9 + 2 x + 6x + 8

7 x + 4x + 4 2

4.

x+1 x + 2x - 15 2

+

x - 4 x -x-6 2

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Complex Fractions

What do you do? There are two ways to do these problems: Method 1: 1. Get a common denominator within the numerator, adjust the numerators, and write the fractions as one fraction. 2. Get a common denominator within the denominator, adjust the numerators, and write the fractions as one fraction. 3. Keep the top fraction, change the operation from division to multiplication, flip the bottom fraction. 4. Factor and simplify, if possible.

Method 2: 1. Find the LCD for all denominators within the complex fraction. 2. Multiply each term by the LCD (if this is done correctly, you will eliminate the fractions within the fraction. 3. Factor and simplify, if possible.

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Examples: 1.

2.

3. 

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Equations and Inequalities

I.  Undefined Fractions When is the fraction undefined? State the  domain using interval notation.

1.  Be sure the fraction is in simplest form and then set the  denominator equal to zero and solve.  Those solutions must be  restricted (removed) from the domain.   Remember for interval notation: Brackets mean included,  parenthesis mean not include, positive and negative infinity can  never be 'included'

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II.  Factoring ­ Discussed Earlier ­  To solve using factoring:  be sure one side of the  equations is equal to zero, factor and then set each factor  equal to zero and solve.  

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III.  Complete the Square Process for Completing the Square 1.  Be sure the quadratic coefficient is 1. 2.  Move constant to the other side of the equation. 3.  Take 1/2 of the linear coefficient, square it, add it to both  sides of the equation. 4.  Rewrite the one side of the equation as a quantity squared,  i.e.  (x ± ?)2 5.  Take the square root of both sides ­ (remember, there is a  positive and a negative square root of a number). 6.  Solve for x. Be sure the radical is in simplest radical form.  

1.  x2 + 8x ­ 3 = 0

2.  2x2 ­ 5x + 1 = 0

3.  2x2 ­ 7x ­ 14 = 0

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Quadratic Formula

Must be sure to have one side equal to zero,  before you solve the equation.   Do not approximate the radical unless you are  told to do so.  

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Solving Algebraic Fractions

1.  Either multiply every term by the LCD (to clear the  fractions) or get a common denominator for every  term, so you can then focus on the numerators only   (since the denominators are all equal now, you may  'drop' the denominators). 2. Solve the resulting equation. 3.  Check your solutions for a possible undefined  fraction.  

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Absolute Value Equations How to solve absolute value equations: 1.  Get the absolute value term alone. 2.  Set what is inside the absolute value sign equal to the other  side of the equation.   Then, set what is inside the absolute value sign equal to the  negation of the other side of the equation.   3.  Solve each equation for x. 4. Check the values for x to make sure they work.

1.  |2x + 5| = 4x + 3

2.

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Inequalities Simple ­Linear

Remember: when you multiply or divide by a negative, you  must switch the direction of the inequality sign. Solve and write the solution using interval notation.

1.   ­3