Context for Mathematics: Algebra & Functions

Question 1

A combination of one or more values arrange in terms that are added together. Could be a single number, including zero.

Answer 1

Expressions

Question 2

In a variable term, there is a variable and this real number value.

Answer 2

Coefficient

Question 3

Numbers without a variable

Answer 3

Constants
Constant Terms

Question 4

Expression that represents the sum of a single variable term, where the variable has no exponent, and a constant, which may be zero.
For example: ax + b

Answer 4

Linear expression

Question 5

On a graph with two points, can be found with the formula (y2 - y1) / (x2 - x1). Represented by the variable, m, or the ratio, rise over run.

Answer 5

Slope

Question 6

Equation that can be written as ax + b = 0, where a is not 0.

Answer 6

Linear Equation

Question 7

Set of all solutions of an equation, can include one, multiple, or zero solutions.

Answer 7

Solution Set

Question 8

A solution set for an equation with no true values.

Answer 8

Empty Set

Question 9

Forms of linear equations.
Hint: there are 5 main types.

Answer 9

Standard Form
Slope Intercept Form
Point-Slope Form
Two-Point Form
Intercept Form

Question 10

Ax + By = C
The slope is -A / B and the y-intercept is C / B.

Answer 10

Standard Form

Question 11

y = mx + b
The slope is m, and b is the y-intercept.

Answer 11

Slope Intercept Form

Question 12

y-y1 = m (x - x1)
The slope is m and (x1, y1) is a point on the line.

Answer 12

Point-Slope Form

Question 13

y-y1 / x-x1 = y2 - y1 / x2 - x1
(x1, y1) and (x2, y2) are two points on the given line.

Answer 13

Two-Point Form

Question 14

x / x1 + y / y1 = 1

(x1, 0) is the point at which the line intersects the x-axis, and (0, y1) is the point at which the same line intersects the y-axis.

Answer 14

Intercept Form

Question 15

A set of simultaneous equations that use the same variables. A solution must be true for each equation.

Answer 15

System of Equations

Question 16

Consistent systems have this many solutions.

Answer 16

At least one solution.

Question 17

Inconsistent systems have this many solutions.

Answer 17

No solutions.

Question 18

Strategies for solving systems of equations equations.
Hint: there are 3 main strategies.

Answer 18

Substitution
Elimination
Graphing

Question 19

The following describes the strategy for finding the…

“x = -b / 2a to find the x-coordinate, then substitute that value back into the equation to find the corresponding y-coordinate;

“a” is the coefficient of x^2 and “b” is the coefficient of x in the standard form of the equation (y = ax^2 + bx + c).”

Answer 19

Vertex of a Parabola

Question 20

The following describes the strategy for finding the…

“Using two points (x1, y1) and (x2, y2), average the x-coordinates, average the y-coordinates.

The formula = (x1 + x2) / 2 , (y1 + y2) / 2.

Answer 20

Midpoint of Two Points

Question 21

The following describes the strategy for finding the…

“Using two points (x1, y1) and (x2, y2) to create the hypotenuse and lines parallel to the x & y axis to create a right triangle. Then use the Pythagorean theorem, a^2 + b^2 = c or c = sqrt (a^2 + b^2).

The formula = sqrt ( (x2 - x1)^2 + (y2 - y1)^2 ).

Answer 21

Distance between Two Points

Question 22

Single constant, variable, or product of constants and variables, such as 7, x, 2x, or xy^3. There will never be addition or subtraction.

Answer 22

Monomial

Question 23

Algebraic expressions that use addition and subtraction to combine two or more monomials.

Answer 23

Polynomials

Question 24

The sum of the exponents of the variables.

Answer 24

Degree of a Monomial

Question 25

The highest degree of any individual term.

Answer 25

Degree of a Polynomial

Question 26

The following describes the strategy for…

“Find the sum of like terms. Terms have the same variable part. Add the coefficients, but keep the variable and exponent the same”

Answer 26

Adding Polynomials

Question 27

The following describes the strategy for…

“Use distributive property to distribute the minus sign. Then, find the sum of like terms. Terms have the same variable part. Add the coefficients, but keep the variable and exponent the same”

Answer 27

Subtracting Polynomials

Question 28

The following describe strategies for…

“FOIL method, Box method and the distributive property.”

Answer 28

Multiplying Polynomials

Question 29

The following describe strategies for…

“Factoring. Long division, written as quotient + (remainder / divisor).”

Answer 29

Dividing Polynomials

Question 30

The following is the ___________ theorem of algebra.

“Every non-constant, single-variable polynomial has exactly as many roots as the polynomial’s highest exponent. For example, if x^4 is hte largest exponent, the polynomial will have exactly 4 roots. Some may be real or complex numbers, or multiplicity (repeats).

Answer 30

Fundamental Theorem of Algebra

Question 31

The following is the ___________ theorem of algebra.

“States that if a polynomial function f(x) is divided by a binomial (x - a), where a is a real number, the remainder of the division will be the value of f(a). If f(a) = 0, then a is the root of the polynomial.”

Answer 31

Remainder Theorem

Question 32

The following is the ___________ theorem of algebra.

“Related to the remainder theorem, states that if f(a) = 0, then (x - a) is a factor of the function.”

Answer 32

Factor Theorem

Question 33

The following is the ___________ theorem of algebra.

“States that any __________ of a polynomial equation with integer coefficients is in the form of (p / q), where p is a factor of the constant term and q is a factor of the leading coefficient.

For example: f(x) = 3x³ - 5x² + 4x + 2.

The constant term is 2 and its factors are ± 1 and ± 2. These would be the values of p.
The leading coefficient is 3 and its factors are ± 1 and ± 3. These would be the values of q.

Values of p/q when q = ± 1:
p/q = ± 1/ ±1, ±2 / ±1 = ± 1, ± 2.

Values of p/q when q = ± 3:
p/q = ± 1/ ±3, ±2 / ±3 = ± 1/3, ± 2/3

Answer: The possible rational zeros of f(x) are ± 1, ± 2, ± 1/3, and ± 2/3.”

Answer 33

Rational Root Theorem

Question 34

The solutions that satisfy the equation when y=0; in other words, where the graph touches the x-axis. Can be found using the quadratic formula, factoring, completing the square, or graphing the function.

Answer 34

Roots

Question 35

The following describes which strategy for solving a quadratic equation.

“Rewrite the equation in standard form, ax^2 + bx + c = 0, where a, b, and c, are coefficients. Substitute into x = (-b +/- sqrt (b^2 - 4ac) / (2a). Evaluate and simplify. Check by substitution.”

Answer 35

Quadratic Formula

Question 36

In a quadratic formula, the portion of the formula under the radical (b^2 - 4ac).

Answer 36

Discriminate

Question 37

In the quadratic formula, if the discriminate (the portion under the radical) is 0, there are _____ roots.

Answer 37

One Root

Question 38

In the quadratic formula, if the discriminate (the portion under the radical) is positive, there are _____ __________ ______ roots.

Answer 38

Two Different Real Roots

Question 39

In the quadratic formula, if the discriminate (the portion under the radical) is negative, there are _____ _________ _______ roots.

Answer 39

Two Complex Roots
(No Real Roots)

Question 40

The following describes which strategy for solving a quadratic equation.

“Begin by rewriting the equation in standard form, ax^2 + bx + c = 0. The goal is to find f and g in (x + f) (x + g) format so that (f + g) = b, and fg = c. Determine the factors of c and look for pairs that could sum to b. Write these as two binomials, (x + f) (x + g), set each to 0, and solve for each x.”

Answer 40

Factoring

Question 41

The following describes which strategy for solving a quadratic equation.

“Manipulate the equation into the form x^2 + bx + c +/- d = 0, so it can be rewritten⁠ using a perfect square, (x + k)^2 + d = 0 and solve for x in the resulting equations.”

Answer 41

Completing the Square

Question 42

The following describes which strategy for solving a quadratic equation.

“Factor the equation and set each factor equal to zero”

Answer 42

Zero Product Property