Special situations in Factoring

Factoring is something we run into all the time in algebra. It is important to know how to do it. Below are the different types of factoring.


Difference of two squares

• a²- b² = (a + b)(a - b)
  • x²- 25 = (x + 5)(x - 5)
    • y²- 49 = (y + 7)(y - 7)
      • z²- 16 = (z + 4)(a - 4)

Trinomial perfect squares

• a² + 2ab + b² = (a + b)(a + b) or (a + b)²
  • a² + 4a+  4= (a+ 2) (a + 2) or(a +2)²
  • a² + 8a + 16  =   (a +4)( a+ 4 )      or (a +4)²
  • a² + 10a+25  = (a + 5 )( a + 5 )or  (a + 5 )²
• ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
  • ^{a2 - 14a + 49 (a - 7)  (a - 7)  or (a - 7)2}
  • ^{a2 - 16a+  64=    (a - 8)   (a -8)   or (a - 8 )2}
  • ^{a2 - 18a+  81    = (a - 9)   (a  - 9 )   or (a - 9)2}

Difference of two cubes

• a³ - b³
  • 3 - cube root 'em
  • 2 - square 'em
  • 1 - multiply and change
    • x3-27
    • x3-1
    • 8y3-125

Sum of two cubes

• a³ + b³ 
  • 3 - cube root 'em
  • 2 - square 'em
  • 1 - multiply and change
  • q3+1
  • a3+125
  • h3+64 

End Behaviors

In an equation, the degree is the highest power. For example, in the equation 5x^8 + 2x^4 - 3x, the degree would be 8. the degree tells the type of equation and the number of turns it's graph has, which is always one number less than the degree.

Degree 0- constant

Degree 1- linear.

Degree 2- quadratic.

Degree 3 – cubic.

Degree 4 – quadratic.

Degree 5- quintic.

Degree 6 –sextic

Degree 7 and on-#tic

Linear Equations

When m is positve: 

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 (Parabolas)

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)

Identifying

A quadratic equation in standard form is  y = ax² + bx + c. In vertex form it is y = a(x - h)² + k.

Here a few good things to remember:

These expressions are usually written in terms of an x, y, or z.

The a cannot be 0.

 

The equation tht makes a parabola is y = a(x - h)² + k where (h, k) is the vertex of the parabola.(fig. 2,3) In both forms, a determines the size and direction of the parabola. The larger the absolute value of a, the steeper (or thinner) the parabola is, since the value of y is increased more quickly. All parabolas are actually similar in a geometric sense, just as all circles and squares are similar figures, with apparent size determined by the constant a.

 

 

If a is positive, the parabola opens upward, if negative, the parabola opens downward.

 

 

 

 

 

 

THE CIRCLE

A circle is the locus of all points, in a plane that is a fixed distance from a fixed point, called the center.

The fixed distance spoken of here is the radius of the circle.

The equation of a circle with its center at the origin is

where (x,y) is a point on the circle and r is the radius (r replaces d in the standard distance formula). Then

 

or

 

The other form of guadratic is a hyperbola. A hyperbola is a plane curve having two branches, formed by the intersection of a plane with both halves of a right circular cone at an angle parallel to the axis of the cone. It is the locus of points for which the difference of the distances from two given points is a constant. the equation is f(x) = 1/x

 

Naming Polynomials

There are many different polynomials. They each have different names oddly enough.
 What follows is a list of the different types.
Degree                                                                  Terms
0 (constant)                                                           monomial
1 (linear)                                                                binomial
2 (quadratic)                                                          trinomial
3 (cubic)                                                                quadrinomial
4 (quartic)                                                              polynomial
5 (quintic)
6 (nth)

Graphing y=a|x-h|+k

y=a|x-h|+k is the equation for absolute value.
The vertex for your graph is (H,K)

The A determines whether your graph will open up or down. If your A is negative it will open down. If your A is positive it will open up. The A will also determine the slope. If A is say, -3, then the graph would have a slope of -3/1 and would open downward.

The H determines the horizontal location of the vertex and the K determines the vertical location. However it is opposite of what would be expected with the horizontal variable in the fact that if H is negative then it moves to the right and if it's positive then it moves to the left.

The green graph is the graph of the new function. It is a translation of the blue graph moved one unit down and 2 units left.

Types of Systems

 When graphing, there are many different equations. First off is the most well known, linear. The equation of a linear line is y=x. This is a consistent/ independent. This equation generally looks like this when graphed:

There are others such as the parent equation y=x^2. When graphed this will form a parabola, a u-shaped line.It is a consistent/ independent graph. Then similar to that is the Absolute Value equation or y=|x|. This ends up as v-shaped line. It is a consistent/ independent graph.

The last three are Square Root equation, Exponential equation, and Rational equation. Square Root is y=rx.  This is a consistent/ independent. It's graph looks like this:

 'r' stands for radical and the symbol looks like this:

Exponential has the equation of y=2^x. This a consistent/ independent line. It looks like this:

Finally rational has an equation of y=1/x. This is an inconsistent graph. It looks like this: