### Review package

• One variable is related to another by a function involving constant terms and terms of 1st order or higher.
• A polynomial with a degree of 2 is Quadratic Relation.
• For constant increments of the independent variable, a relation is quadratic if the second differences of the dependent variable are constant.
• A graph that is either concave upwards or concave downwards
• When the rate of change is constant it is a linear relation
• When the rate of change is not constant then its not a linear relation

## FACTORING

• Factoring is finding what to multiply to get an expression
• To factor a number means breaking it into an equation where a number can be multiplied to get the original

There are many different types of factoring I will explain all.

## Common Factoring

Firstly we have - Common Factoring

• If all the terms in the equation are divisible by the same number or variable then that term is called the common factor
• A polynomial is not completely factored if the GCF (greatest common factor) is not factored out.
• Expanding and factoring are 2 completely different things a(b+c) <= (factoring) ab+bc <= (Expanding)
Examples

1. 25x+50
=> 25(x+2)

=> (factored Fully because 25 is the GCF)

2. 25x+50

=> 5(5x+10)

=> (It is NOT factored fully as 5 and 10 still have a common factor)

## Grouping

factoring by grouping is factoring in done by grouping pairs of terms. Then factor each pair by groups with ()

Examples

x^2+2x^2+8x+16

=> x^2(x+2) +8(x+2)

=> (x+2) (x^2+8)

## SIMPLE TRINOMIALS

• Simple trinomial equations are in the form of, x^2+bx+c
• In solving trinomials, the brackets have to be expanded (Multiply each term in the first bracket to each term in the second bracket).
• Equations in the form of x^2+bx+c are called Quadratic Equations

Expand=> x^2+bx+c , (x+2) (x+1)

• (x+2) (x+1)
= x^2+3x+2
• "c" value is the product. Find out 2 numbers that add up to the "b" value and when multiplied the answer is the "c" value.
Examples:

1. x^2 +7x -30 [Product=-30 , Sum=7]
= (x+10) (x-3)

## COMPLEX TRINOMIALS

COMPLEX TRINOMIALS

• Factor by Decomposition.
• Complex Trinomial are in the form of, Ax^2 +Bx +C.
• The "C" value is multiplied by the A value, only the coefficient.
• Determine the factors of the product which also add up to the "B" value.

Factor by Decomposition: Ax^2 +Bx +C

• 2x^2 +3x -5 [Product: A*C= -5*2=-10 , Sum=3]
=2x^2 +5x-2x -5 [Decompose the Middle term]
=x(2x+5) -1(2x+5)
=(x-1) (2x+5)
Examples:
1. 5x^2 +10x+2x+4
=5x(x+2) +2(x+2)
=(5x+2) (x+2)

## FACTORING DIFFERENCE OF SQUARES

FACTORING DIFFERENCE OF SQUARES

• The "a" value is squared (x^2)
• In the form of, (a+b) (a-b)
• Square-root of "b" value

Expand:

• (x+2) (x-2)

=x^2 -2x+2x -4
=x^2 - 4
• x^2 -16
= (x^2 - 4)
Example:

4x^2 - 22
=2(2x^2 - 11)

## COMPLETING THE SQUARE

• Used to change Standard form equations to Vertex form

STEPS TO COMPLETE THE SQUARES:

1. Block off the first two terms
2. Factor out the A value only
3. Divide the middle term by 2 and square it (b/2)^2
5. Take out the negative squared value, Multiply the "a" value by the negative squared value when taking it out of the bracket
6. Write within brackets x + the square root of the number and square the entire bracket and solve what is outside the brackets
Example:

1. x^2 + 6x - 2
=(x^2 + 6x) - 2
=(x^2 + 6x + 9 - 9) -2
=(x^2 + 6x + 9) -9 - 2
=(x+3)^2 - 11

• In the form of, ax^2 + bx + c = 0
• Formula is used to identify the "x" value
• Quadratic formula is used when the equation cannot be factored
• Quadratic formula give 2 values of "x"

x = -b ± √b^2 - 4ac /2a

EXAMPLES:

1. x^2 - x + 8 = 0
x= -b ± √b^2 - 4ac /2a
x= 4x^2 + 33x + 8 = 0
x= -33 ± √33^2 - 4(4)(8) / 2(4)
x= -33 ± √961 / 8
x= -33 ± 31 / 8
x= -1/4 x= -8

THE END HOPE IT HELPED!!!!!!!!!!!!!!