Identifying Special Situations in Factoring

Difference of two squares

-a²- b² = (a + b)(a - b)

Trinomial perfect squares

--a² + 2ab + b²= (a + b)(a + b) or (a + b)²

^{--a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}

Difference of two cubes

--a³ - b³

3 - cube root 'em

2 - square 'em

1 - multiply and change

Sum of two cubes

--a³ + b³ 

3 - cube root 'em

2 - square 'em

1 - multiply and change

Binomial expansion

--(a + b)³ 

--(a + b)⁴

 

Naming Polynomials and End Behaviors

Linear Equations:
y=mx+b (1 degree, 0 turns)
m: slope
b: y-intercept

Domain= x-values
Range= y-values

When m is positive: falls to the left, rises to the right

domain → +∞, range → +∞

domain → -∞, range → -∞

When m is negative: rises to the left, falls to the right

domain → -∞, range → +∞

domain → +∞, range → -∞

Quadratic Equations:
 --Parabolic Equations
y=ax² (2 degree, 1 turn)

(a+b)(c+d)

When a is positive: rises to the left, rises to the right

domain → +∞, range → +∞ (rises on the right)

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

When a is negative: falls to the left, falls to the right

domain → +∞, range → -∞ 

domain → -∞, range → -∞

***Number of turns is always one less than the degree!

Degree:

0- Constant 

1- Linear

2- Quadratic 

3- Cubic

4- Quartic

5- Quintic 

6 to ∞- nth Degree 

Terms:

Monomial 

Binomial 

Trinomial 

Quadrinomial 

Polynomial

Identifying Quadratic Equations

STANDARD FORM: ax² + bx + cy² + dy + e= 0 

If a is not equal to c and the signs are the same then the equation is an ellipse:

If a or c have different signs then the equation is a hyberbola:

If a or c=0, then the equation is a parabola:

If a=c, then the equation is a circle:

How to Multiply Matrices

-In order to multiply matrices, you have to first write a dimension statement. This statement states that the columns of the first matrix must match the rows of the other matrix in order for them to be able to be multiplied.

Example of a dimensions statement: 3 x 2and2 x 3

 

These two matrices could not be multiplied because the columns and rows do not match up.



                                               



 

 

 

 

Dimensions of a Matrix

-- The dimension of a matrix consist of the rows and columns that make it up.
          -Columns: vertical
          -Rows: horizontal

--When figuring out what the dimension of a matrix is you ALWAYS put the row first, then the columns.

This matrix has a dimension of 1x3 because it has one row(horizontal) and 3 columns(vertical)

This matrix has a dimension of 3x3 because it has 3 rows and 3 columns

This matrix has a dimension of 2x3 because it has 2 rows and 3 columns

This dimension is known as the Indentity Matrix (serves as the mulitplicative identity. This is also known as a 3x3 matrix since it has 3 rows and 3 columns.

Error Analysis

For this problem the x values are going up by 5 which would make the slope 2, not 10. After plotting the point on a graph the solution you get is not equal to y= 9 + 10x which is why this problem is wrong. The right equation would be y= 2x + 9.

mber 20 the line should be dotted because the y is only greater than the equation, not greater than or equal to. For number 21 it's wrong because the y is less than or equal to the equation which means it should be shaded below the line, not above.

For number 22 the y is only less than x +3 which would mean that it wouldn't be a solid line, but a dotted line and for number 23 the line is correct but the shading would need to be on the opposite side of the line since it's greater than, not less than.

After pluggin in the solution (1, -2) to both of these equations it shows that this solution works for the first equation but not the second one and in order for this point to be the correct solution in would have to work both of the problems.

Graphing y=a|x-h|+k

-The vertex of this equation in (h,k)

-a tells you whether the V opens up or down on the graph.

           -- you always go up and right then up and left, or down and right then down and left.

- the h determines whether the V moves right or left ( since it's the absolute values of h you use the opposite of that number)

-the k determines whether the V moves up or down.