Identifying Different Situations In Factoring

Identifying special situations in factoring

• Difference of two squares
• a²- b² = (a + b)(a - b)
• (x + 9)(x − 9)
• (6x − 1)(6x + 1)
• (x³ − 8)(x³ + 8)
• Trinomial perfect squares
  • a² + 2ab + b²= (a + b)(a + b) or (a + b)²
    • 16x² - 8xy + y² = (4x - y)²
      
      
    • 8xy + y² + 16x² = 16x² + 8xy + y² = (4x + y)²
    • 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
      • x³-27 = (x-3)(x²+3x+9)
      • 8y³-125 = (2y-5)(2y²+10y+25)
      • s³-1 = (s-1)(s²+s+1)
• 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+125 = (a+5)(a²-5a+25)
      • h³+64 = (h+4)(h²-4h+16)
• Binomial expansion
• (a + b)³ = (a+b)(a+b)(a+b)
• (a + b)⁴ = (a+b)(a+b)(a+b)(a+b)

End Behaviors/ Naming Polynomials

Linear Equations:

y= mx+b

1 degree

0 turns

Domain - x values
Range - y values referred to as f(x)

When m is Positive:

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

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

When m is Negative

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

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

Quadratic Equations (parabolic equation)

y=ax²

2 degree

1 turn

(a+b)(c+d)

When a is Positive

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

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

When a is Negative

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

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

Naming Polynomials:

--Number of turns is always 1 less than the degree.

Degree:

0- Constant

1- Linear

2- Quadratic

3- Cubic

4- Quartic

5- Quintic

6 to ∞- nth Degree

Terms:

Monomial - one term

Binomial - two terms

Trinomial - three terms

Quadrinomial - four terms

Polynomial - two or more terms

Quadratic Equations

How to identify quadratic equations:
ax² + bx + cy² + dy + e= 0

4x² + 4y² = 36 is a circle because a=c. The a is 4 and the c is 4.

2x² + 4y = 3 is a parabola because a or c equals 0.

4x² - 4y² = 12 is a hyperbola because a and c have different signs.

4x² + 3y² = 25 is an ellipse because a is not equal to c, and the signs are the same.

Can you multiply these matrices?

In trying to determine if you can multiply a set of matrices, you would have to write a Dimensions Statement first.

2x2 times 2x2 = 2x2

Example :

2x2 times 2x2 = 2x2

If the numbers on both sides of the " times" word are equal, then you are able to multiply the matrices. The two outside numbers determines the size of your answer. Which in this equation it is "2x2."

If you are able to multiply the matrices, then you should multiply row by column and then add the sum of the products to get the answer.

Scalar Multiplication

Scalar multiplication is when you distribute the outside number to all the other numbers in the bracket.

Dimensions of a Matrix

Dimensions of a matrix are determined by looking at the rows and columns.

The columns are the numbers that go up or down (vertical) . The rows are the numbers that go across (horizontal) . The dimension, once again, is found by

Row X Column ( Horizontal X Vertical) .

This matrix has a dimension of 1 x 3 since it has ONE row and THREE columns.

This matrix has a dimension of 3 x 3 because it has THREE rows and THREE columns.

This is called the identity matrix, where 1's are in the diagonal line. This is called the identity matrix because if you multiply this matrix with another matrix, the answer will be the same as the other matrix.

Error Analysis

The equation stated above is incorrect. This is also not in the slope intercept form, y=mx+b.

9= 9+10(0) 19=9+10(5) ... etc, does not equal out.

Therefore, the correct equation is y=2x+9. 9=2(0)+9 , 19=2(5)+9. The equation works out.

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In question number 22, the error is that the line should be dotted instead of a solid line because it's only "less than", NOT " less than OR EQUAL TO. "

In question number 23, the error is that the shaded area should be on the other side of the line because it is GREATER than or equal to.

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The student didn't check for the other equation, x+4y=-5.

1+4(-2)=-5. The solution does not equal -5. So it does not work.

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In question number 20, the lines should be dashed instead of solid because the equation is LESS THAN, not equal to.

In question number 21, the shaded region should be below the lines because it is LESS THAN or equal to.

Graphing Absolute Values (y=a|x-h|+k)

• The equation for absolute value is y=ax-h+k.
• The vertex can be represented by (h,k).
• The "a" variable determines if the graph opens up or opens down.
• The "h" variable shifts the graph either to left or to the right.
• The "k" variable shifts the graph either up or down.

Types of Systems

• Consistent/ Independent- Consistent and independent graphs have lines that intersect, have exactly one solution which is (x,y), and they also have different slopes.
• Consistent/ Dependent- Consistent and dependent graphs have lines with an infinite amount of solutions. They also have the same slope and y- intercept.
• Inconsistent- Inconsistent graphs have no solutions. They have parallel lines but different y- intercepts.