Polynomial Equations

A polynomial equation is one that can be written in the form of:
axⁿ + bx^n-1 + . . . + ax + m = 0

a, b, a and m are constants.
the biggest exponent of x showing in a non-zero term of a polynomial the degree of that certain polynomial.

 6x + 1= 0 has degree 1, since the largest power of x that occurs is x = x¹. Degree 1 equations are called linear equations.

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.

 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.

 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:

1: Degree 0- constant.

2: Degree 1- linear.

3: Degree 2 – quadratic.

4: Degree 3 – cubic.

5: Degree 4 – quadratic.

6: Degree 5- quintic.
7: Degree 6 –sextic.

8: Degree 7 - degree with number terms.

Quadratic Functions

stardard form is ax² + bx + cy² + dy + e= 0.

if your equation looks like 2x² + 2y² = 4 , then it is a circle because a = c.

if c or a = 0, the equation is a parabola.

if a and c have different signs, the equation is a hyperbola

if a is not equal to to c, but they have the same signs, then the graph would look like a ellipse.

Multiplying Matrix

to determine how you need to multipy the matrices you must write a demension statement.
[5,6] [5,2,1]
[3,8] [2,6,9]
[6,8]

this demension would be 3 x 2 times 2 x 3.

the underlined number must be the same for the problem to be able to solve it.
the numbers on the outside need to be the same as well.
the equation for this is :

Demensions of a matrix

this is a 1 x 3 matrix because there is 1 row and 3 columns.

this is a 3 x 3 because there are 3 rows and 3 columns.

Error Analysis

in problem 20, the line should be dotted.
in problem 21, the shading should be below the lines.

in problem 22, the line should be dotted because it is not equal to.
in problem 23, the shading should be above the line, no below it.

absolute value equations

the equation is:
y = a | x-h | + k

Isolate the absolute value
If the number on the other side of the equation negative, then it is no solution.
If the number on the other side of the equation is not negative, write two equations without absolute value bars, but have one where the number the equation equals to, be negative, and the other postive.
Then solve both equations.

|3x - 6| - 9 = -3

|3x - 6| = 6

3x - 6 = 6          3x - 6 = -6

3x = 12       3x = 0

x = 2          x = 0

Graphing absolute value equations:

the parent function for graphing absolute values is y = | x |

If the equation was y = | x + 2 | , this is how it would look:

If there was a negative outside of the absolute value sign, then it would open downwards, and look something like this:    y = - | x |

systems of linerar equations

Independent Consistant:
a equation is consistant if there is at least one solution.
a equation is independent when 2 lines cross at one point

for example:
y = x + 10
y = 2x

Dependent Consistant:
when 2 lines have the same slope and y-intercept.
one overlaps the other

for example :
y = x + 10
2y = 2x + 20



 
 
Incosistant :
this means the 2 lines do not cross at any point.
they are most likely parallel.
they do have the same slope.
 
example:
y = x
y = x + 10