GRE - Algebra
Multiply (x + 2)(3x - 7)
3x^2 - x - 14
Simplify 15y^2 - 9y
3y(5y - 3)
How many terms does the following algebraic expression have: w^3z + 5z^2 - z^2 + 6
4
How many terms does the expression 4x^6 + 7x^5 - 3x + 2 have? What are the coefficients and degrees of each term? What is the degree of the polynomial?
4 Coefficient of 4 and degree of 6 Coefficient of 7 and degree of 5 Coefficient of -3 and degree of 1 Constant term with degree of 0 6
Simplify 4x + 12
4(x + 3)
What is (3^2)(3^4)?
729
Simplify 2x + 5x
7x
For all values of x for which the expression is defined, what is (7x^2 + 14x) / (2x + 4) equivalent to?
7x/2
What is (3/4)^2?
9/16
For all integers a and b and all positive numbers x, except x = 1, what property holds true?
If x^a = x^b, then a = b
What is the first rule of solving linear inequalities?
When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original
Simplify the algebraic expression (x^2 - 9) / (4x - 12). For what domain?
(x + 3) / 4 All x that is not equal to 3
What is Rule 6 of exponents?
(x/y)^a = x^a / y^a
What is Rule 3 of exponents?
(x^a) / (x^b) = x^(a - b) = 1 / (x^(b-a))
What is Rule 2 of exponents?
(x^a)(x^b) = x^(a + b)
What is Rule 5 of exponents?
(x^a)(y^a) = (xy)^a
What is Rule 7 of exponents?
(x^a)^b = x^(ab)
What is the second rule of solving linear inequalities?
When both sides of the inequality are multiplied or divided by the same non-zero constant, the direction of the inequality is preserved if the constant is positive, but the direction is reversed if the constant is negative
When is a fraction not defined?
When the denominator is equal to 0
What are the three steps in simplifying (7x^2 + 14x) / (2x + 4)
1. Factor the numerator and denominator 2. Cancel out the common terms
What can the domain of the function be (2)?
1. Given explicitly and restricted to a specific set of values of x 2. All values for x
What is possible regarding linear equations (2)?
1. No solutions 2. Identity (true for all values of x)
What are the two basic methods for solving systems of linear equations?
1. Substitution 2. Elimination
What is 4^(-3)?
1/64
What is (5^7) / (5^4)?
125
How many terms does the following algebraic expression have: y -(1/4)
2
What is/are the solution(s) of 2x^2 - x - 6 = 0
2 and -3/2
What is the denominator of (7x^2 + 14x) / (2x + 4)? When is this equal to 0? What is the original expression defined for?
2(x+2) When x = -2 All x that is not equal to -2
What is (2^3)(3^3)?
216
Simplify 3xy + 2x - xy - 3x
2xy - x
If 2^(3c + 1) = 2^10, what is c?
3
How many terms does the expression 2x^2 - 7xy^3 - 5 have? What are the coefficients and degrees of each term? What is the degree of the polynomial?
3 Coefficient of 2 and degree of 2 Coefficient of -7 and degree of 4 Constant term with degree of 0 4
Translate the following verbal description into an algebraic expression: If the square of the number x is multiplied by 3 and then 10 is added to that product
3x^2 + 10
How many real solutions can quadratic equations have?
0, 1 or 2
How many terms does the following algebraic expression have: 2x
1
How many terms does the following algebraic expression have: 8 / (n+p)
1
What is 7^0?
1
What degree does a variable that is written without an exponent have? What is the degree of a constant term?
1 0
What is (2^5)^2?
1,024
How can an algebraic expression be written (2)?
1. As a single term 2. As a sum of terms
What is a variable?
A letter that represents a quantity whose value is unknown
What is an inequality?
A mathematical statement that uses <, >, ≤, or ≥
What is a coefficient?
A number that is multiplied by variables
What is an ordered pair?
A pair of numbers (x, y)
What is a system of equations?
A set of equations in two or more variables
What is an identity?
A statement of equality between two algebraic expressions that is true for all possible values of the variables involved
What is an equation?
A statement of equality between two mathematical expressions
What is a constant term?
A term that has no variable
In linear equations in two variables, what do these systems have? Explain. What is also possible (2)?
A unique solution Only one ordered pair of numbers that satisfies both equations in the system 1. No solutions 2. Infinitely many solutions
What is the domain of h(x) = |x|
All real numbers
What is the domain of f(x) = 2x / (x - 6)
All real numbers except for 6
What is a linear equation?
An equation involving 1+ variables in which each term in the equation is either a constant term or a variable multiplied by a coefficient
What is an algebraic expression?
An expression that has one or more variables
What is the solution of linear equations in two variables?
An ordered pair of numbers that makes the equation true
In the algebraic expression x^a, what is x? What is a?
Base Exponent
In algebraic expressions, how can like terms be combined?
By adding their coefficients
How can some quadratic equations be solved more quickly?
By factoring
What is one way to find solution of a quadratic equation? What is it?
By using the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a
What can happen to a number or variable that is a factor of each term in an algebraic expression?
Can be factored out
What can an algebraic expression in one variable be used to do?
Define a function of that variable
How are two algebraic expressions multiplied?
Each term of the first expression is multiplied by each term of the second expression and the results are added
For any function, what does each input x give? However, what can more than one value of x give?
Exactly one output The same output
What does every linear equation in two variables have?
Infinitely many solutions
What is/are the solution(s) of x^2 + x + 5 = 0
No real solutions
What happens in the substitution method?
One equation is manipulated to express one variable in terms of the others
What is a quadratic polynomial?
Polynomials of degree 2
What is a cubic polynomial?
Polynomials of degree 3
What are like terms?
Terms that have the same variables raised to the same exponents
What is the first rule important to producing equivalent equations?
When the same constant is added to or subtracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original
What are simultaneous equations?
The equations in the system of equations
What is the degree of a polynomial?
The greatest degree of its terms
What is the domain of a function?
The set of all permissible values of the variable x
What is the solution set of an inequality?
The set of all values of the variable that make the inequality true
What is a polynomial?
The sum of a finite number of terms in which each term is either a constant term or a product of a coefficient and 1+ variables with positive integer exponents
What is the degree of a term?
The sum of exponents of the variables in the term
What is f(x)? How is it obtained?
The value of f at x By substituting the value of x into the expression
What are the solutions?
The values of the variables that make the equation true
What does it mean to solve a system of equations in two variables, x and y?
To find ordered pairs of numbers (x, y) that satisfy all of the equations in the system
What does it mean to solve a system of equations in three variables, x, y and z?
To find ordered triples of numbers (x, y, z) that satisfy all the equations in the system
What does it mean to solve an equation?
To find the values of the variables that make the equation true
What is the object of the elimination method?
To make the coefficients of one variable the same in both equations so that one variable can be elimination via addition or subtraction
What are equivalent equations?
Two equations that have the same solutions
What are equivalent inequalities?
Two inequalities that have the same solution set
Generally, what do systems of linear equations in two variables consist of? What does each contain?
Two linear equations One or both of the variables
What is the third rule important to producing equivalent equations?
When an expression that occurs in an equation is replaced by an equivalent expression, the equality is preserved and the new equation is equivalent to the original
What is the second rule important to producing equivalent equations?
When both sides of an equation are multiplied or divided by the same non-zero constant, the equality is preserved and the new equation is equivalent to the original
How can a linear equation in two variables, x and y, be written? What are a, b and c? What are a and b not equal to?
ax + by = c Real numbers 0
What is the equation for the quadratic equation?
ax^2 + bx + c = 0
Simplify w^3z + 5z^2 - z^2 + 6
w^3z + 4z^2 + 6
Solve the inequality ((4x + 9) / 11) < 5
x < 11.5
What is/are the solution(s) of x^2 + 4x + 4 = 0
x = -2
What is/are the solution(s) of 2x^2 - x - 6 = 0 using factoring
x = -3/2, 2
What is/are the solution(s) of 5x^2 + 3x - 2 = 0 using factoring
x = 2/5, -1
What is the solution to 11x - 4 - 8x = 2(x + 4) - 2x
x = 4
Use elimination to solve: 4x + 3y = 13 x + 2y = 2
x = 4 y = -1
Use substitution to solve: 4x + 3y = 13 x + 2y = 2
x = 4 y = -1
What is the domain of g(x) = x^3 + √(x + 2) - 10
x ≥ -2
Solve the inequality -3x + 5 ≤ 17
x ≥ -4
What is Rule 1 of exponents?
x^(-a) = 1/(x^a)
What is Rule 4 of exponents?
x^0 = 1