### Review Package

• Sometimes a curve of best fit is more appropriate model for data than a line of best fit.
• For constant increments of the independent variable (x), a relation is quadratic if the second differences of the dependent variable (y) constant.
Example: In a table of values the first differences are not constant but the second differences are constant.
• A polynomial of degree 2 models a quadratic relation.
• A linear relation models a phenomenon where the rate of change is constant.
• A non-linear relation models a phenomenon where variable is the rate of change.

## Factoring

COMMON FACTOR

• Factoring is the opposite of expanding, ab+ac = a(b+c)
• If every term of a polynomial is divisible by the same constant, the constant is called a common factor.
• A polynomial is not considered to be completely factored until the Great Common Factor (GCF) has been factored out.

Factor: ab+ac
= a(b+c) , 4x+20
= (x+5)

• 4x+20
= 2(2x+10) (Not completely Factored)
• 4x+20
= 4(x+5) (Completely Factored)

EXAMPLES:

1. 3x-15y
= 3(x+5y)
2. 4a^3b^4 - 6a^2b^2 + 2ab
= 2ab(2a^2b^3 - 3ab+1)

___________________________________________________________________________________________

FACTOR BY GROUPING

• This is done by grouping a pair of terms. Then, factor each pair of two terms.
• x^2 + 2x^2 + 8x+16
= x^2(x+2) +8(x+2)
= (x+2) (x^2+8)
• 8am-3bn-6an+4bm
= 4m(2a-b) -3n(b+2a)

___________________________________________________________________________________________

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 +6 [Product=6 , Sum=7]
= (x+1) (x+6)
2. x^2 +7x -30 [Product=-30 , Sum=7]
= (x+10) (x-3)
3. x^2 +3x +5 [Product=5 , Sum=3]
=Not Factor-able

*Note: In some cases, first you might have to factor out.
Example:

Q: -5x^4 +5x^3 +30x^2 [First factor and then solve as a Simple Trinomial]
=-5x^2(x^2-x-6) [Product=-6 , Sum=-1]

=-5x^2(x-3)(x+2)

___________________________________________________________________________________________

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. 14x^2 -19x -3 [Product: A*C= 14*-3=-42 , Sum=-19]
=5x^2 +10x+2x +4 [Decompose the Middle term]
=5x(x+2) +2(x+2)
=(5x+2) (x+2)
2. x^2 +7x +12 [Product: A*C= 12*1=12 , Sum=7]
=x^2 +3x+4x +12 [Decompose the Middle term]
=x(x+3) +4(x+3)
=(x+1) (x+3)
3. 5x^2 +10x+2x+4
=5x(x+2) +2(x+2)
=(5x+2) (x+2)

___________________________________________________________________________________________

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)

EXAMPLES:

1. (a-3) (a+3)
=a^2 +3a-3a -9
=a^2 - 9
2. (p-5) (p+5)
=p^2 - 25
3. 4x^2 - 22
=2(2x^2 - 11)

Factoring binomials - difference of squares: Quick Explanation!

## COMPLETING THE SQUARES

• 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

EXAMPLES:

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
2. 3x^2 + 24x - 11
=3(x^2 + 8x) - 11
=3(x^2 + 8x + 16 -16) - 11
=3(x^2 + 8x +16) -48 -11
=3(x^2+4)^2 - 59

Completing the Square to Solve Quadratic Equations: More Examples - 1

• 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
2. 24= x^2 - x +30
x^2 - x + 6 = 0
x= -b ± √b^2 - 4ac /2a
x= -(-1) ± √(-)^2 - 4(1)(6) / 2(1)
x= 1± √-23 / 2
x= No Solution