7.1 the greatest common factor and factoring by grouping finding the greatest common factor:...

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


• Example: factor 5x2y + 25xy2z

)5(5255

55

525

55

22

0111

12122

01212

yzxxyzxyyx

xyzyxGCF

zyxzxy

zyxyx


• 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


• 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)


• 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


• 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


• 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


• Box Method (not in book):

6?

2

?2

?)?)(2(672

2

2

xx

x

xxxx


• 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


• Perfect square trinomials:

• Examples:

222

222

2

2

yxyxyx

yxyxyx

2222

2222

15152511025

333296

zzzzz

mmmmm


• Difference of 2 cubes:

• Example:

2233 yxyxyxyx

)933327 2333 wwwww


• Sum of 2 cubes:

• Example:

2233 yxyxyxyx

)933327 2333 wwwww


• 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


• 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


• 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


• 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


• 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


• 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)


• 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)


• 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.

7.1 the greatest common factor and factoring by grouping finding the greatest common factor:...

Documents

factoring trinomials

bx c factoring x

bx slide

bx c factoring ax

factoring binomials

bx c example

greatest common factor

negative slide