Post on 17-Dec-2015
7.1 The Greatest Common Factor and Factoring by Grouping
• Finding the Greatest Common Factor:1. Factor – write each number in factored form.2. List common factors3. Choose the smallest exponents – for variables
and prime factors4. Multiply the primes and variables from step 3
• Always factor out the GCF first when factoring an expression
7.1 The Greatest Common Factor and Factoring by Grouping
• Example: factor 5x2y + 25xy2z
)5(5255
55
525
55
22
0111
12122
01212
yzxxyzxyyx
xyzyxGCF
zyxzxy
zyxyx
7.1 The Greatest Common Factor and Factoring by Grouping
• Factoring by grouping1. Group Terms – collect the terms in 2 groups
that have a common factor2. Factor within groups3. Factor the entire polynomial – factor out a
common binomial factor from step 24. If necessary rearrange terms – if step 3 didn’t
work, repeat steps 2 & 3 until you get 2 binomial factors
7.1 The Greatest Common Factor and Factoring by Grouping
• Example:
This arrangement doesn’t work.
• Rearrange and try again
)815()65(2
815121022
22
xyyx
xyxyyx
)32)(45(
)23(4)32(5
8121510 22
yxyx
xyyyxx
xyyxyx
7.2 Factoring Trinomials of the Form x2 + bx + c
• Factoring x2 + bx + c (no “ax2” term yet)Find 2 integers: product is c and sum is b
1. Both integers are positive if b and c are positive
2. Both integers are negative if c is positive and b is negative
3. One integer is positive and one is negative if c is negative
7.2 Factoring Trinomials of the Form x2 + bx + c
• Example:
• Example:
)1)(4(
414 ;514
452
xx
xx
)3)(7(
21)3(7 ;437
2142
xx
xx
7.3 Factoring Trinomials of the Form ax2 + bx + c
• Factoring ax2 + bx + c by grouping 1. Multiply a times c
2. Find a factorization of the number from step 1 that also adds up to b
3. Split bx into these two factors multiplied by x
4. Factor by grouping (always works)
7.3 Factoring Trinomials of the Form ax2 + bx + c
• Example:
• Split up and factor by grouping
62014
)6(20)8(15
)4(30)2(60120
15148 2
b
ac
xx
)52)(34(
)52(3)52(4
15620815148 22
xx
xxx
xxxxx
7.3 Factoring Trinomials of the Form ax2 + bx + c
• Factoring ax2 + bx + c by using FOIL (in reverse)
1. The first terms must give a product of ax2
(pick two)2. The last terms must have a product of c (pick
two)3. Check to see if the sum of the outer and inner
products equals bx4. Repeat steps 1-3 until step 3 gives a sum = bx
7.3 Factoring Trinomials of the Form ax2 + bx + c
• Example:
correct 672)2)(32(try
incorrect 682)1)(62(try
incorrect 6132)6)(12(try
?)?)(2(672
2
2
2
2
xxxx
xxxx
xxxx
xxxx
7.3 Factoring Trinomials of the Form ax2 + bx + c
• Box Method (not in book):
6?
2
?2
?)?)(2(672
2
2
xx
x
xxxx
7.3 Factoring Trinomials of the Form ax2 + bx + c
• Box Method – keep guessing until cross-product terms add up to the middle value
)2)(32(672 so
642
32
32
2
2
xxxx
x
xxx
x
7.4 Factoring Binomials and Perfect Square Trinomials
• Difference of 2 squares:
• Example:
• Note: the sum of 2 squares (x2 + y2) cannot be factored.
yxyxyx 22
wwww 3339 222
7.4 Factoring Binomials and Perfect Square Trinomials
• Perfect square trinomials:
• Examples:
222
222
2
2
yxyxyx
yxyxyx
2222
2222
15152511025
333296
zzzzz
mmmmm
7.4 Factoring Binomials and Perfect Square Trinomials
• Difference of 2 cubes:
• Example:
2233 yxyxyxyx
)933327 2333 wwwww
7.4 Factoring Binomials and Perfect Square Trinomials
• Sum of 2 cubes:
• Example:
2233 yxyxyxyx
)933327 2333 wwwww
7.4 Factoring Binomials and Perfect Square Trinomials
• Summary of Factoring1. Factor out the greatest common factor
2. Count the terms:
– 4 terms: try to factor by grouping
– 3 terms: check for perfect square trinomial. If not a perfect square, use general factoring methods
– 2 terms: check for difference of 2 squares, difference of 2 cubes, or sum of 2 cubes
3. Can any factors be factored further?
7.5 Solving Quadratic Equations by Factoring
• Quadratic Equation:
• Zero-Factor Property:If a and b are real numbers and if ab=0then either a = 0 or b = 0
02 cbxax
7.5 Solving Quadratic Equations by Factoring
• Solving a Quadratic Equation by factoring1. Write in standard form – all terms on one side
of equal sign and zero on the other
2. Factor (completely)
3. Set all factors equal to zero and solve the resulting equations
4. (if time available) check your answers in the original equation
7.5 Solving Quadratic Equations by Factoring
• Example:
1,5.2 :solutions
01or 052
0)1)(52( :factored
0572 :form standard
7522
2
xx
xx
xx
xx
xx
7.6 Applications of Quadratic Equations
• This section covers applications in which quadratic formulas arise.
Example: Pythagorean theorem for right triangles (see next slide)
222 cba
7.6 Applications of Quadratic Equations
• Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2
a
b
c
7.6 Applications of Quadratic Equations
• Examplex
x+1
x+2
3
0)1)(3(
032
4412
)2()1(
2
222
222
x
xx
xx
xxxxx
xxx
9.3 Linear Inequalities in Two Variables
• A linear inequality in two variables can be written as:
where A, B, and C are real numbers andA and B are not zero
CByAxCByAx
CByAxCByAx
or
or or
9.3 Linear Inequalities in Two Variables
• Graphing a linear inequality:1. Draw the graph of the boundary line.
2. Choose a test point that is not on the line.3. If the test point satisfies the inequality, shade
the side it is on, otherwise shade the opposite side.
or for dashedit Make
or for solidit Make
9.4 Systems of Linear Equations in Three Variables
• Linear system of equation in 3 variables:
• Example:LKzJyIx
HGzFyEx
DCzByAx
3232
1437
284
zyx
zyx
zyx
9.4 Systems of Linear Equations in Three Variables
• Graphs of linear systems in 3 variables:1. Single point (3 planes intersect at a point)
2. Line (3 planes intersect at a line)
3. No solution (all 3 equations are parallel planes)
4. Plane (all 3 equations are the same plane)
9.4 Systems of Linear Equations in Three Variables
• Solving linear systems in 3 variables:
1. Eliminate a variable using any 2 equations
2. Eliminate the same variable using 2 other equations
3. Eliminate a different variable from the equations obtained from (1) and (2)
9.4 Systems of Linear Equations in Three Variables
• Solving linear systems in 3 variables:
4. Use the solution from (3) to substitute into 2 of the equations. Eliminate one variable to find a second value.
5. Use the values of the 2 variables to find the value of the third variable.
6. Check the solution in all original equations.