Identifying special situations in factoring

• Difference of two squares
• a²- b² = (a + b)(a - b)
• 3 examples
• Trinomial perfect squares
• a² + 2ab + b² = (a + b)(a + b) or (a + b)²
• 3 examples
• ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
• 3 examples
• Difference of two cubes
• a³ - b³
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• 3 examples
• Sum of two cubes
• a³ + b³ 
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
• 3 examples
• Binomial expansion
• (a + b)³ = Use the pattern
• (a + b)⁴ = Use the pattern

degrees and polynomials

A polynomial equation is an equation that can be written in the form

axⁿ + bx^n-1 + . . . + rx + s = 0,

where a, b, . . . , r and s are constants.
We call the largest exponent of x appearing in a non-zero term of a polynomial the degree of that polynomial.

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Examples
1. 3x + 1 = 0 has degree 1, since the largest power of x that occurs is x = x¹. Degree 1 equations are called linear equations.
2. x² - x - 1 = 0 has degree 2, since the largest power of x that occurs is x². Degree 2 equations are also called quadratic equations, or just quadratics.
3. x³ = 2x² + 1 is a degree 3 polynomial (or cubic) in disguise. It can be rewritten as x³ - 2x² - 1 = 0, which is in the standard form for a degree 3 equation.
4. x⁴ - x = 0 has degree 4. It is called a quartic.

The polynomials name is depend on their degree of exponents. There many names of in the polynomials. The names are following below:

Name 1: Degree 0- constant.

Name 2: Degree 1- linear.

Name 3: Degree 2 – quadratic.

Name 4: Degree 3 – cubic.

Name 5: Degree 4 – quadratic.

Name 6: Degree 5- quintic.

Name 7: Degree 6 –sex-tic.

Name 8: Degree 7 - degree with number terms.

Linear Equations: 

y= mx+b 
1 degree
0 turns 

When m is Positive: 
 Rises to the Right, Falls to the left. 

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

When m is Negative 
Rises to the Left, Falls to the Right 

• 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 
Rises Left, Rises 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 → -∞ (falls on the right)
• domain → -∞, range → -∞ (falls on the left)

Naming Polynomials: 
--Number of turns is always 1 less than the degree

Identifying Circles, Parabolas, Hyperbolas, and Ellipses

4x²+4y²=36   Circle

2x²+4y=3     Parabola

4x²-4y²=12  Hyperbola

4x²+3y²=25  Ellipse

A=C: Circle

A≠C and same sign: Ellipse

A and C have different signs: Hyperbola

A or C are 0: Parabola

Matrices

An m×n matrix can be multiplied by an n×p matrix, and the result is an m×p matrix. The number of columns in the first matrix must be the same as the number of rows in the second matrix. For example, a 4 ×2 matrix can be multiplied by a 2 ×3 matrix to produce a 4 ×3 matrix.

Error Analysis

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.

 (1,2) is a solution for the system 5x-y=7, but it is not a solution for the other system x+4y= -5.

Absolute Value Function

Absolute Value equations are written as y=a|x-h|+k . To find the vertex of the graph , h and k will be your vertex , which looks like a coordinate pair : (h,k) . In the equation , A tells whether the V opens up or down . H moves the V right or left (opposite), and K moves V up and down.

Systems of Equations

No solution

Inconsistent

elemination equations