Identifying Special Situations in Factoring

• Difference of two squares
• a²- b² = (a + b)(a - b)
• a²- 9 = (a + 3)(a - 3)
• a²- 25 = (a + 5)(a - 5)
• a²- 4 = (a + 2)(a - 2)
• Trinomial perfect squares
• a² + 2ab + b²= (a + b)(a + b) or (a + b)²
• a² + 8a + 16= (a + 4)(a + 4) or (a +4)²
• a² + 12a + 36= (a + 6)(a + 6) or (a + 6)²
• a² + 10+25= (a + 5)(a + 5) or (a + 5)²
• ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
• a^{2 _} 6a + 9 = (a - 3)(a - 3) or (a - 3)²
• a^{2 _} 10a + 25 = (a - 5)(a - 5) or (a - 5)²
• a^{2 _} 8a + 16 = (a - 4)(a - 4) or (a - 4)²
• Difference of two cubes
• a³ - b³
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• a³ - 1 = (a - 1)(a² + a + 1
• a³ - 27 = (a - 3)(a² + 3a + 6)
• a³ - 64 = (a - 4)(a² + 4a + 16)
• Sum of two cubes
• a³ + b³ 
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• a³ + 1 = (a + 1)(a² - a + 1)
• a³ + 64 = (a + 4)(a² - 4a + 16)
• a³ + 125 = (a + 5)(a² - 5a + 25)
• Binomial Expansion
• (a + b)³ = a³ + 3a²b + 3ab² + b³= (a + 4) ³ = a³ + 12a² + 48a + 64
• (a + b)⁴ =a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ = (a + 3) ⁴  = a⁴  + 12a³ + 54a² + 108a + 81

                        

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

Examples:
5x - 2 = 0 is a linear binomial
with a degree of 1 and no turns. 
5x³ − 4x² + 7x − 8= 0 is a cubic quadrinomial 
with a degree of 3 and 2 turns.
x⁴ - x² + 4x - 24 = 0 is a quartic quadrinomial
with a degree of 4 and 3 turns.

Number of Turns is always 1 less than the degree

Linear Equations : Y=mx+b, 1 degree and 0 turns

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

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

Quadratic Equations: ax² + bx + c = 0, 2 degree and 1 turn  

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

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

 

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

How to Identify Quadratic Functions?

Standard Form of a Quadratic

ax² + bx + cy² + dy + e = c




If a=c, then the equation is a circle.
Example- 2a² + 2c² = 24

If a does not equal c and the signs are the same then the equation is an ellipse.

Example- 9a² + c² = 3

If a or c = 0, the equation is a parabola.

Example- a ²+a-3 = 0

If a and c are different signs the equation is a hyperbola.

Example- 3a² - 3c² = 36

Can you multiply Matrices?

To multiply matrices we have to:

1. We have to write a dimension statement to see whether or not the matrix can be multiplied.
2. If it can be multiplied then we multiply ROW x COLUMN.
3. Then get the sum of the product.

Dimension Statement Example:
[2   3] [-6]
[4   7] [-8]

Dimension Statement: 2 x 2 times 2 x 1

The numbers in green tell you whether or not the matrix can be multiplied, in this case they can be multiplied because the numbers are the same.
The numbers in blue tell you the size of the final answer.

Dimensions of a Matrix

• The matrix has 1 row and 3 columns. So the dimensions of this matrix is 1 x 3.

• 
  
   The matrix has 3 rows and 3 columns. So the dimension of this matrix is 3 x 3. Also called a Square matrix.

   





• The matrix has 3 rows and 2 columns. So the dimensions of this matrix is 3 x 2.

• The matrix has 3 rows and 3 columns. So the dimension of this matrix is 3 x 3. This matrix is an identity matrix. 

Error Analysis

The slope of this table should be 2, because the x values are increasing by 5. So the equation for this table would be y=2x+9.

The student only checked one equation to see if the solution is right. The student should check both equations. And the solution only solves 5x-y=7, not x+4y=-5.

For number 23, the line of the graph should be a dotted line instead of a solid line. For number 24, the graph should be shaded above the line instead of below because y is greater. 

The line on number 20 should be dotted instead of a solid line. The graph on number 21 should be shaded below because y is greater than or equal to.

Graphing Y=a|x-h|+k

When graphing y=a|x-h|+k, the a is an indicator of the slope of the graph. The h of the graph tells us whether we move the graph to the right or left. For example if the equation says x+h, then the graph moves to the left because it is opposite of the equation present. The k tells us whether the graph will move up or down. For example if the equation says plus k, then the graph will move up; therefore, h and k is the vertex of the graph.

Systems of Equations

Consistent/Independent- system in which two lines cross at one point; one solution. Diffrent slope.

Inconsistent - has no solution; parallel lines.

Consistent/Dependent- has infinitely many solutions; SAME line, SAME slope.