ALGEBRA REVIEW

FINDING REAL ROOTS

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COMMON FACTOR POLYNOMIAL

If all terms of a polynomial have a common factor, it can be factored out: 4x²-6x+2= 2(2x²-3+1)

ADDING OR SUBTRACTING POLYNOMIALS

Only add or subtract the like terms: (3x+2) + (4x²+x+2) = 4x² + 2x + 4

EQUATIONS WITH |x|

Solve two versions of the equation, one with being positive and one the other negative: |ax+b|=c ax+b = c OR -(ax+b) = c

RADICAL

The symbol for root is called a radical √

MULTIPLYING MONOMIALS To multiply monomials ___ the coefficients and ___ the powers.

multiply the coefficients and add the powers

EVALUATING EXPRESSIONS 1) When you evaluate an expression, you ____, leaving no ____.

plug in numbers for the variables, leaving no variables in any form.

ROOTS OF A POLYNOMIAL***

...

COEFFICIENT 1) A coefficient is? 2) T/F: A coefficient cannot change its value

1) A constant multiplier of a variable. 2) True

TERM 1) A term is 2) Like terms are 3) There cannot be any ___ or___ within a single term 4) 3xy is / is not the same as 3yx.

1) A product of coefficients and any number of variables (with or without exponents) are called terms: example: 3x+5 has two terms "3x" and "5" 2) Terms that have the same variables with identical exponents for each variable are called "like terms": ex: 3x²y and 7x²y are "like terms" because they have the same variables x and y with identical exponents. 3) Addition or subtraction 4) IS

ADDING \ SUBTRACTING MONOMIALS 1) Adding and subtracting monomials is done by 2) Can only be added or subtracted if

1) Adding or subtracting the coefficients. 2) they have the same variables with identical powers.

ADDING RADICALS 1) Two terms of same variable with the same fractional power can be aadded together by ___. 2) Examples

1) Adding their coefficients. 2) 2√x + 3√x = 5√x; x^(³/₄) + 5x^(³/⁴) = 6x(³/⁴)

MONOMIALS 1) Define 2) Examples of monomials 3) Examples of "not monomials"

1) An expression with only one term is a monomial. 2) 3x; 2x²,4xy are monomials 3) 2x+1; x+y are NOT monomials

BINOMIALS 1) Define

1) An expression with only two terms: 3x+2 or 3x²+4xy

VARIABLE 1) A variable is 2) T/F: A variable can have more than one value

1) An unknown quantity whose value may change 2) True

POLYNOMIALS 1) What are polynomials? Examples? 2) What is the order of a polynomial? Are monomials or binomials polynomials?

1) Any expression with one or more terms is called a polynomial. Ex: 3x+2, 4x+3xy+y² are all polynomials. 2) The order of the polynomial is the highest power of any variable. 3) Yes. They are polynomials of order 1 and 2.

SIMPLIFYING RADICAL EXPRESSIONS 1) A radical expression can be simplified by 2) (x²+3x+2) / (x+2) = (x+2)(x+1)/ (x+2) =x+1

1) Cancelling any common factor between numerator and denominator

DIVIDING RADICALS 1) To divide a term with the same variable and fraactional powers... 2) Examples:

1) Divide the coefficients and subtract the exponents. 2) 2x^²/³ ÷ 4x^¼ = 2/4x^(2/3 -1/4) = 1/2x^(⁵/¹²)

SYSTEMS OF LINEAR EQUATIONS 1) Given two independent linear equations with the same two variables, we can solve for each variable by 2) Example: Solve ax+by=c and dx+ey=f

1) Eliminating the other using arithmetic operations: adding or subtracting the same number from both sides, and mulitplying or dividing both sides by the same number Examples: 1. Mulitply first by d and second by a: adx+bdy=cd ; and adx+aey=af 2. Subtract first from second: aey-bdy=af-cd 3. Solve for y:y(ae-bd)=af-cd ---> y=af-cd / ab-bd 4. Use y to solve for x.

SYSTEMS OF INEQUALITIES 1) RULES

1) Express each inequatlity in terms of y. 2) Draw the line and shade the appropriate regions (example: shade above for > shade below for <). 3) The final region is the intersection of all shaded areas.

MULTIPLYING POLYNOMIALS

1) Open up the first polynomial and multiply the second polynomial by each term: (3x+2) x (4x²-x+2) = 12x³+5x² +x +4

RADICAL EXPRESSION 1) An expression that contains a fractional power of any variable is called a 2)

1) Radical Expression. 2) √+2x, ³√x+y, ⁵√5+3

SUBTRACTING RADICALS 1) Two terms of same variable with the same fractional power can be aadded together by ___. 2) Examples

1) Subtracting their coefficients. 2) 2√x - 3√x = -1√x; x^(³/₄) - 5x^(³/⁴) = -4x(³/⁴)

ALGEBRAIC FRACTION 1) An algebraic fraction is? 2) An algebraic fraction can be simplified by? 3) Remember to? 4) Examples:

1) The division of one polynomial by another. 2) Cancelling the common factors between the numerator and the denominator. 3) Factor out the numerator and the denominator to find the common factors. 4) (x²-a²) ÷ (x-a) = (x+a)(x-a)÷(x-a) = (x+a)

DOMAIN OF A FUNCTION 1) A function f(x) need not be defined for all values of x. The set of values of x for which the function is defined is called the ___ 2) Examples: 2a) f(x) = 3x+2 is defines for ___ 2b) f(x) = √x is defined ____ 3) A function f(x)is not defined for values of x that involve division by 4) For the GRE, a function f(x) is not defined for values of x that involve square roots of

1) The domain of the function. Basically, all the values of x that you can put into a function and still get a viable answer. 2a) is defined for all real values of x 2b) is defined for all x≥0 3) Zero 4) Negative numbers

ASSOCIATIVE LAW 1) Define

1) The grouping does not affect the product or sum of 3 or more algebraic terms 2x²+(4y+1) = (2x²+4y)+1

COMMUTATIVE LAW 1) Define 2) What is not commutative?

1) The order does not affect the product or sum of algebraic terms. 2) Subtraction or division.

LINEAR INEQUALITIES: 1) To Solve a Linear Inequality: 2) When dividing or multiplying by a negative number,

1) To solve a linear inequalities with one variable, isolate the variable to one side of the arithmetic operation. 2) Change the direction of the sign. Same is true for ≥, and ≤

FUNCTIONS 1) f(x) is a function of the variable x if for each x there is a unique number f(x). Function is a rule that matches or maps each number x to a 2a) f(x)=3x is a function that 2b) f(x)=x² is a function that 2c) f(x)=1 is a function that

1) Unique number F(x). 2a) maps each number x to three times its value 3x 2b) maps each number x to its square x² 2c) maps each number x to the same number, 1

DIVIDING A POLYNOMIAL BY A MONOMIAL 1) How do you do this? 2) if the denominator is a higher order polynomial ___

1) Use distributive law, separate the numerator and reduce each term: 4x²-6 / 2x = 4x²/2x - 6x/2x = 2x-3 2) Do not distribute the denominator.

MULTIPLYING BINOMIALS 1) How do you multiply binomials?

1) Use the FOIL method. (simply the distributive law)

DISTRIBUTIVE LAW 1) Define

1) When multiplying a sum or difference of terms, the distributive property of multiplication allows us to distribute the multiplying term among the terms being added or subtracted.

LINEAR EQUATION of ONE VARIABLE 1) A linear equation of one variable is____. 2) Linear equations have no ___. 3) All linear equations can be written as: 4) The solution of linear equations can be written as

1) a linear equation of one variable where the variable has unit power. 2) ino squares, square roots, or other powers. 3) ax +b = c 4) x = (c-b) ÷ a

EXPRESSION: SUBSTITUTION 1) Define 2) After substitution, a variable....

1) a method where one or more variables in an expression is replaced by numbers or another set of variables. 2) does not appear anywhere else in the expression

ADDING AND SUBTRACTING BINOMIALS 1) To add or subtract binomials, ___ the ___ of the like terms

1) add or subtract the coefficients of the like terms.

QUADRATIC FORMULA 1) All quadratic equations of one variable can be written as 2) You can use the quadratic formula to solve 3) The quadratic formua is:

1) ax²+bx+c=0 2) all quadratic equations 3) x = -b ± √b²-4ac) / 2a This will give you two answers for X. For example: X = 2, OR x=-5

CONSTANT TERMS 1) A constant term is a term that 2) Constant terms sit alone and does not

1) does not contain any variable. ex: in 3x+5, 5 is the constant term. 2) multiply any variable or function of a variable.

SOLVING QUADRATIC EQUATIONS 1) Solving quadratic equations of the form ax²+bx+c=0, first 2)NOTE: All quadratic equations of ne variable can be written as:

1) factor left side example: x²+3x+2=0 --> (x+2)(x+1)=0; This means x=-2 or x=-1 2) ax²+bx+c=0

EQUATION: IN TERMS OF: 1) Given an equation of two variables, x and y, we can express x in terms of y by? 2) All linear equation of two variables can be written as?

1) isolating x to the left side of the equation, and y to the right. 2) ax+by+c=d... Therefore, x=(d-by-c)/a

ALGEBRAIC EXPRESSION 1) An algebraic expression is 2) There is no ____ in an algebraic expression

1) one or many terms connected by algebraic operations. examples: (3x+5), (xy+a), (xy+a)÷(x+y) 2) equals sign

RANGE OF A FUNCTION 1) Define

1) the range of a function is the set of values a function f(x) can produce for each x value. Thus, in f(x) = y, the range is all y values that can be produced for the various values of x that can be inputted into the system.

FRACTIONAL POWER 1) When a number has a fractional power...

1) when the number of a fractional power m/n as the exponent, the numerator m acts as the integer power and the denominator n as the root: 2∧(3/2) = 2³∧¹/²

FACTORING (x²-y²) 1) The differences between squares can be factored as ____ ** This appears frequently in the GRE

1) x²-y² = (x+y)(x-y)

THE N-TH ROOT 1) y is the n-th root of a numer x for a positive integer n if: 2) The n-th root of a number x is expressed as 3) Square roots and cube roots are examples of n-th roots where n=2 and n=3, respectively.

1) yⁿ=x 2) x∧(¹⁺ⁿ) 3) 16∧(¹/₄) =2

CUBE Root

For GRE assume all cube roots are positive.