# CSET Multiple Subjects: Subtest 2a, Domain 2

## 47 terms · CSET Multiple Subjects Subtest 2: Mathematics Domain 2: Algebra

P = 4a
A = a²

P = 2b + 2h
P = 2(b + h)
A = bh

P = 2a + 2b
P = 2(a + b)
A = bh

P = a + b + c
A = (bh)/2

P = 4a
A = ah

### Trapezoid

P = b1 + b2 + x + y
A = [h(b1 + b2)]/2

C = πd
C = 2πr
A = πr²

SA = 6a²
V = a³

### Rectangular Prism

SA = 2(lw + lh + wh)
SA = (Base-per)h + 2(Base-area)
V = lwh
V = (Base-area)h

### Prisms in general

SA = (Base-per)h + 2(Base-area)
V = (Base-area)h

### Cylinder

SA = (Base-per)h + 2(Base-area)
SA = 2πrh + 2πr²
SA = 2πr(h + r)
V = (Base-area)h
V = πr²h

SA = 4πr²
V = (4/3)πr³

### Evaluating expressions

Insert the value(s) given for the unknown(s) and do the arithmetic, making sure to follow the rules for the order of operations.

Example: Evaluate 2x² - 4y + 11 if x = 3 and y = -5.
2(3)² - 4(-5)+ 11
2(9) - (-20) + 11
18 + 20 + 11
38 + 11
49

### Equation

-A relationship between numbers and/or symbols that says two expressions have the same value
-Solving an equation for a variable requires that you find a value or an expression that has the desired variable on one side of the equation and everything else on the other side of the equation
-By doing the same arithmetic to each side of the equation, you eventually can isolate the desired variable

Example: x-5 = 23. Solve for x.
Add 5 to each side of the equation
x = 28
Replace the original x with 28 and check to see if the resulting sentence is true.
28 - 5 = 23
23 = 23

### Proportion

-A statement that says that two expressions written in fraction form are equal to one another.
-Proportions are quickly solved using a cross multiplying technique

Example: Solve for x. 3/x = 5/7
5x = 21
x = 21/5 or 4 1/5

### Similar triangles

Have corresponding sides forming proportions

### Inequality

-A statement in which the relationships are not equal
-Instead of using an equal sign (=) as in an equation, we use > (greater than) and < (less than), or ≥ (greater than or equal to) and ≤ (less than or equal to).
-When working with inequalities, treat them exactly like equations, EXCEPT: If you multiply or divide both sides by a negative number, you must reverse the direction of the sign.

Example: Solve for x: -7x > 14
Divide by -7 and reverse the sign
x < -2

### Monomial

An algebraic expression that consists of only one term

Examples:
9x
4a²
3mpxz²

### Polynomial

An algebraic expression that consists of two or more terms separated with either addition or subtraction

Examples:
x + y (a polynomial with two terms)
x² + 3x - 4 (a polynomial with three terms)

Must be like terms (like terms have exactly the same variables with exactly the same exponents on them)

Example: 5x and 7x are like terms, but 5x and 7x² are not like terms

Example: Simplify the following: 17a + 7b - 12a - 10b
Rewrite with like terms near each other
17a - 12a + 7b - 10b
5a - 3b

Add or subtract the like terms in the polynomials together

Example: (3x² - 7x + 12) + (5x² + 9x - 19) = ?
(3x² - 7x + 12) + 5x² + 9x - 19
3x² + 5x² - 7x + 9x + 12 - 19
(3 + 5)x² + (-7 + 9)x + (12 - 19)
8x² - 2x - 7

### Multiplying monomials

When an expression has a positive integer exponent, it indicates repeated multiplication
(Multiply numbers and add exponents on like term variables)

Example: (7x²)(-5x³)
(7 · -5)(x² · x³)
-35x^5

### Multiplying monomials with polynomials and polynomials with polynomials

Use the distributive property

Example: (3x +5)(2x - 7)
First distribute the 3x over the (2x - 7), and then distribute the +5 over the (2x - 7)
3x(2x - 7) + 5(2x - 7)
6x² - 21x + 10x - 35
6x² - 11x - 35
This also means that 6x² - 11x - 35 = (3x + 5)(2x - 7)

### Factoring

Finding two or more quantities whose product equals the original quantity

### Factoring out a common factor

-Find the largest common monomial factor of each term
-Divide the original polynomial by this factor to obtain the second factor (the second factor will be a polynomial)

Example: Factor completely 2y³ - 6y
2y³ - 6y = 2y(y² -3)

### Factoring the difference between two squares

-Find the square root of the first term and the square root of the second term
-Express your answer as the product of the sum of the quantities from step 1 times the difference of those quantities

Example: x² - 144
x² - 144 = (x + 12)(x - 12)

### Factoring polynomials that have three terms: Ax² + Bx + C

-Check to see if you can monomial factor (factor out common terms). Then, if A = 1 (the first term is simply x²), use double parentheses and factor the first term. Place these factors in the left sides of the parenthesis. For example, (x )(x )
-Factor the last term, and place the factors in the right side of the parentheses
To decide on the signs of the numbers, do the following. If the sign of the last term is negative:
-Find two numbers whose product is the last term and whose difference is the coefficient (number in front) of the middle term
-Give the larger of the two numbers the sign of the middle term, and give the opposite sign to the other factor
If the sign of the last term is positive:
-Find two numbers whose product is the last term and whose sum is the coefficient of the middle term
-Give both factors the sign of the middle term
If A ≠ 1 (if the first term has a coefficient different than 1 — for example, 4x² + 5x + 1), then additional trial and error will be necessary

A quadratic equation is an equation that could be written as Ax² + Bx + C = 0. To solve a quadratic equation:
-Put all terms on one side of the equal sign, leaving zero on the other side
-Factor
-Set each factor equal to zero
-Solve each of these equations

Example: Solve for x. x² - 6x = 16
x² - 6x - 16 = 0
(x - 8)(x + 2) = 0
x - 8 = 0 --> x = 8
or x + 2 = 0 --> x = -2

A quadratic with a term missing

Example: Solve for x. x² - 16 = 0
Factoring, (x + 4)(x - 4) = 0
x + 4 = 0 --> x = -4
or x - 4 = 0 --> x = 4

### Coordinate graphs

-Formed by two perpendicular number lines (coordinate axes)

### Horizontal axis

-x-axis or abscissa
-Numbers to the right of 0 are positive and to the left of 0 are negative

### Vertical axis

-y-axis or ordinate
-Numbers above 0 are positive and numbers below 0 are negative

### Origin

-The point at which the two axes intersect
-Represented by the coordinates (0,0), often marked simply 0

### Coordinates/ordered pairs

-An ordered pair of numbers by which each point on a coordinate graph is located
-Coordinates show the points' location on the graph
-Shown as (x,y)

### x-coordinate

-The first number in the ordered pair
-Shows how far to the right or left of 0 the point is

### y-coordinate

-The second number in the ordered pair
-Shows how far up or down the point is from 0

Four quarters that the coordinate graph is divided into
-In quadrant I, x is always positive and y is always positive
-In quadrant II, x is always negative and y is always negative
-In quadrant III, x is always negative and y is always negative
-In quadrant IV, x is always positive and y is always negative

### Graphing equations

Graphs of equations in two variables (usually x and y) can be formed by finding ordered pairs that make the equation true, and then connecting these points

Example: Make a graph of the equation 2x + y = 6
One way to do this is to set up a table of values with the x-values first, and then the y-values. You then replace one of the variables with values and find what the other variable would have to be for each replacement. If the x's were replaced with -2, -1, 0, 1, and 2, then find the corresponding y-values
If x = -2, then 2(-2) + y = 6 --> -4 + y = 6 --> y = 10
Make a table, plot the points

### Linear equation

-An equation whose points, when connected, form a line
-Can be written in the form, "y = mx + b"

Example: Rewriting 2x + y = 6 in the "y = mx + b" form, you get y = -2x + 6

### y-intercept

-The point at which the line passes through the y-axis
-The b in the y = mx + b form

Example: y = -2x + 6 --> b = 6
Also notice that in the graph, the line passes through the y-axis at y = 6

### Slope value

-The slope of a line gives a number value that describes its steepness and the direction in which it slants
-Positive slope, negative slope, zero slope, undefined/no slope
-Slope is calculated by comparing the rise (the difference of the y-values) to the run (the difference of the x-values), when going from one point to another
-The m in the y = mx + b

Example: Using the points (-2,10) and (-1,8), we can calculate the slope the following way:
Slope = rise/run = (10-8)/[-2 - (-1)] = 2/-1 = -2
Regardless of which two points were chosen, the slope value would be the same
Line falls as it goes to the right, indicating that the slope is negative

### Positive slope

Line rises as it goes to the right

### Negative slope

Line falls as it goes to the right

### Zero slope

Line is horizontal

Line is vertical

### Slope of parallel lines

Same slope values

### Slope of perpendicular lines

Slope values will be opposite reciprocals

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