Identifying special situations in factoring

• Difference of two squares
• a²- b² = (a + b)(a - b)
• a²- 4 = (a + 2)(a - 2)
• a²- 64 = (a + 8)(a - 8)
• a²- 9 = (a + 3)(a - 3)
• Trinomial perfect squares
• a² + 2ab + b²= (a + b)(a + b) or (a + b)²
• a² + 2a + 4= (a + 2)(a + 2) or (a +2)²
• a² + 8a + 16= (a + 4)(a + 4) or (a + 4)²
• a² + 6+9= (a + 3)(a + 3) or (a + 3)² 
• Difference of two cubes
• a³ - b³
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• a³ - 2 = (a - 2)(a² + a + 2)
• a³ - 64 = (a - 3)(a² + 4a + 16)
• Sum of two cubes
• a³ + b³ 
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• q³+1 = (q+1)(q²+q+1)
• a³+100 = (a+10)(a²-10a+5)
• Binomial Expansion
• (a + b)³ = (x+y)³ = x³ + 6x²y + 6xy² + y³
• (a + b)⁴ = (x+y)⁴ = x⁴ + 8x³y + 10x²y² + 8xy³ + y⁴

End Behaviors / Naming Polynomials

Domain - x values

Range - y values referred to as f(x)

 

Naming Polynomials
Degree	Terms
0--Constant	Monomial
1—Linear	Binomial
2—Quadratic	Trinomial
3—Cubic	Quadrinomial
4—Quartic	Polynomial
5—Quintic
6--nth

 

• domain → +∞, range → +∞ (rises on the right)
• domain → -∞, range → -∞ (falls on the left)

• domain → -∞, range → +∞ (rises on the left)
• domain → +∞, range → -∞ (falls on the right)

• domain → +∞, range → +∞ (rises on the right)
• domain → -∞, range → -∞ (falls on the left)

• domain → +∞, range → -∞ (falls on the right)
• domain → -∞, range → -∞ (falls on the left)

Quadratic Functions

Identifying Quadratic Equations

standard form: ax² + bx + cy² + dy + e= 0

A = C : equation of a circle.

A or C = 0 : equation of a parabola.

A or C have different signs : equation of a hyperbola.

A is not equal to C but is the same sign : equation of an ellipse. 

Example:

Equation:

x² + y² = 16

Example: 2 (Ellipse)

√(36-4x²)

-√(36-4x²)

Multiplying Matrices

When multiplying matrices by a scaler, each element is multiplied by the scaler.

For Example:

To multiply matrices by eachother, multiply the elements of the first matrix by the corresponding element in the second matrix.

 

For Example:

To make sure you can multiply 2 matrices together, the inside numbers should be the same when you put the two dimensions next to each other like:

3 x 4  && 4 x 6

Dimensions of A Matrix

Matrices are identified Row X Column

The Matrix above is 3x3.

Count the rows and columns, show how to identify the dimensions

1 x 3

3 x 2

3 x 3

Error Analysis Examples:

X. In number 22, the line should not be solid, but dotted because

it is not "equal to" the given equation.

X. The shading in the line is wrong because it should be shaded

above the line, instead of below it.

X. This analysis is wrong because the given solution does not work

in the second equation. To find the solution, the student should use substitution

or elimination to figure out the solution.

X. This problem is wrong because if you use the given equation, y=9+10x, it does not match with the table above. To find the right equation, graph the table given and find the slope, then write an equation. The answer is:  y = 2x + 9.

X. Number 20 is wrong because the line is supposed to be dotted because it is no equal to the given equation.

X. Number 21 is supposed to be shaded below the given line because it is y is lesser than or equal to the given equation, not greater than.

Graphing y=A|x-h|+k

(h,k) of the equation y=A|x-h|+k is the vertex.

'h' moves the graph on the x intercept while 'k' moves it on the y intercept.
h = horizontal movement , y = vertical movement

Example:

Types Of Systems

01.  Inconsistent Systems has no solutions and never intersect. They are parallel. (same slope, different y-int)

02. Consistent & Independent Systems have one solution and they intersect.

03. Consistent & Dependent Systems have an infinite amount of solutions and they are the same line (same slope & y-int)