Identifying Special Situtations in Factoring

• Difference of two squares
• a²- b² = (a + b)(a - b)
• x^2-4^2 = (x+4)(x-4)
• p^2-8^2 = (x+8)(x-8)
• 11^2-y^2 = (11+y)(11-y)

Trinomial perfect squares

• a² + 2ab + b²= (a + b)(a + b) or (a + b)²
1. 36+12x+x^2 = (6+x)^2
2. 2x^2+13x+21 = (2x+7)(x+3) 
3. 4x^2+y^2+4xy = (2x+y)^2
• ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
• 16x^2+8xy+y^2 = (4x+y)^2
• y^2-8y+16 = (y-4)^2
• 16x^2+24xy+9y^2 = (4x+3y)^2

Difference of two cubes

• a³ - b³
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• s^3-1 = (s-1)(s^2+s+1)
• x^3-27 = (x-3)(x^2+3x+9)
• 8y^3-125

Sum of two cubes

• a³ + b³ 
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change 
•  s^3-1 = (s-1)(s^2+s+1)
• x^3-27 = (x-3)(x^2+3x+9)
• 8y^3-125 = (2y-5)

Binomial expansion

• (a + b)³ = a^3+3a^2b+3ab^2+b^3
• (a + b)⁴ = a^4+4a^3b+6a^2b^2+4ab^3+b^4

End Behaviors

Domain- X-Values (x --> +∞)

Range- Y Values (y --> -∞)

 

• 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)

Identifying Quadratic Equations and Circles

The Standard Form of a Circle: (x-h)^2+(y-k)^2=r^2

The Standard Form of a Quadratic Equation: ax^2+bx+cy^2+dy=e

To find out whether an equation is a Circle, Hyperbola, Ellipse, or a Parabola use the equation: Ax^2+Cy^2=E.

If A=C, the equation is a CIRCLE.
If A does NOT equal C and has the same sign, the equation is an ELLIPSE.
If A=C, but has different signs, the equation is a HYPERBOLA.
If A or C are 0, the equation is a PARABOLA.

Can you multiply these matrices?

In order to know if you can multiply a couple matrices , the number of columns in the first matrix must equal the number of rows in the second matrix. So yes , you can multiply these matrices.



You can multiply these matrices too . The number of columns in matrix one equal the number of rows in the second.

Dimensions of a Matrix

3x3 : 3 rows and 3 columns

3x2 : 3 rows and 2 columns

3x3 : 3 rows and 3 columns

1x3 : 1 row and 3 columns

To figure out the dimensions in a matrix , go by Row x Column . The number of horizontal rows x the number of vertical columns.

Systems of Equations

In Graph 3, the system of equation is an Inconsistent equation . It's lines are parallel and parallel lines will never cross , so there could never be a solution. Inconsistent and No Solution.

In Graphs 1 and 4, two lines are crossing at one point , therefore it only has one solution . This system of equation will be called Independent and Consistent.

In Graph 2, the line looks to be one line , but it is actually two lines drawn together. These two lines intersect at every point along their length. This System of Equation would be called Dependent , and the solution would be the whole line.

The graphing of the inequality in number 20 is correct , but the line should be dotted , not solid. The graphing in number 21 is also correct , but the shaded should be below the graph.