Identifying Special Situations in Factoring

• Difference of two squares
  • a²- b² = (a + b)(a - b)
    • a²-81 ^{2 = (a + 9)( a- 9)}
    • a²- 4^{2 =( a + 2)(a - 2)}
    • a²- 36^{2 = ( a + 6)(a-6)} 
• Trinomial perfect squares
  • a² + 2ab + b²= (a + b)(a + b) or (a + b)²
    • a² + 6a + 9²= (a + 3)(a + 3) or (a + 3)²
    • ^{a2 + 14a + 492= (a + 7)(a + 7) or (a + 7)2}
    • ^{a2 + 18a + 812= (a + 9)(a + 9) or (a + 9)2}
  • ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
    • a² - 8ab + 16² = (a - 4)(a - 4) or (a - 4)²
    • ^{a2 - 6a + 92 = (a - 3)(a - 3) or (a - 3)2}
    • ^{a2 - 12a + 362 = (a - 6)(a - 6) or (a - 6)2}
• Difference of two cubes
  • a³ - b³
    • 3 - cube root 'em
    • 2 - square 'em
    • 1 - multiply and change
      • a³ - 1 ^{= (a-1)(a+a+1)}
      • ^{a3 - 27 = (a-3)(a-3a+9)} 
      • ^{a3 - 125 =(a-5)(a+5a+25)}
• 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 + 125 =} (a+5)(a²+5a+25)
      • ^{a3 + 64 =}  (a+4)(a²+4a+16)
• 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 of Polynomials 2

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

  
  

  

  
  

  Negative linear equations

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

  
  

  

  
  

  Positive linear equations

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

  
  

  
  

  
  

  
  

How to Identify Quadratic Equations

The standard form of a quadratic equation is: ax² + bx + cy ² + dy + e = 0

If a is equal to c in the above equation then the equation will make a circle. 2x ² + 2y² = 24; a = c so the when graphed it would be a circle.

 

If in the equation a dose NOT equal c the the equation is one for an elipse.

-4x² + -25y² = 205, because a dose not equal c but they have the same sign the equation is one of an elipse.

If a or c equals zero then the equation belongs to a parabola.

4x + 2y² = 61 in this equation a = zero so the equation is a parabola.

if a and c have different signsthen the equation is one of a hyperbola. -3x² + 3y² = 36: a and c have different signs so the the shape ia a hyperbola.

multiplying matricies

To begin multplying matrices you must first make a  deminsion statement. a deminsion stateent is a statemaent that states the deminsions of the two matrices you are multiplying in a certian format.

The deminsion statement is 2x3 times 3x1 you can multiply these matrixes together because the columns of the first matrix matches the rows of the second. the numbers that are not used the 2 and the 1 tell you the deminsions of your anwer.


To actually multiply you take the first row and multiply by first column and add the resulting products, to get the number to go into column one row one. You will do this process of each space you have to fill alternating the rows and columns. 


 

dimensions of a matrix

The i,agge above is a matrix. The dimensions of this matrix is 3X3. this is also a square matrix. A square matrix is a matrix whose dimensions contain the same number, such as 3x3, 4x4, or 6x6. '

This matrix is an idenity matrix. When you multiply an inverse matrix by the idenity matrix the resulting answer will be zero. A way to recongize and idenity matxis is by the 1's crossing the matrix on the diaongal.

some more examples of matries dimensions are shown below:

 The dimmensions of this matrix is 3x2.

   

Absolute value functions

y = a | x - h | + k , this is the formula for a absolute value graph. the parent graph (y = |x|) looks like this:

if you wish to move the graph to the left or the right you manipulate the h value, since in the formula the h is negative you move the graph opposite of the sign, as shown below.

to move the graph up and down the y-axis you change the k value.

To strech and shrink the graph you change the a value, if a is a fraction then the graph is vertically streched, and if the graph is vertically shrinked then the coeficcient is a whole number.

To invert the graph you make a negative.

 

Error Analysis

To correct this problem you need to first figure out the slope using the point slope formula: Y1 - Y2 / X1- X2 you should get 10/5 which would simplify to two. the final equation will be y = 2x + 9.

The mistake this student made was not to check the point in both equations. you can fix this problem by using either the elimination or substitute method. 

In the first graph the shading is correct but the line should be dotted instead of solid because the graph is not an or equal to inequality. The second graph's line is correct but the shading is not, to correct the student should shade above the line, not below.

    

The first graph's line needs to be dotted and in the second graph the shading needs to be below the graph.