FACTORING INTRO – PART 1 NAME: _________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

Coach Joseph has 12 girls on her basketball team. How many ways can she group them for practice drills? List the options.

The same idea is true for polynomials. Sometimes, we need the product and sometimes we need the factors.

EX 1: A rectangle has side lengths of 2 x−1 and x+2. These are the factors.

When would we need to use the side lengths?

*Find the product (2 x−1)( x+2)=¿ _______________________

Why would we need the product? What does it mean to the rectangle?

What if you know the area (product) and need the side lengths (factors)? Can we work backwards to find them?

EX 2: What are the side lengths? How do you know?

EX 3: What are the side lengths? How do you know?

Factors

________ groups of _________

________ groups of _________

________ groups of _________

________ groups of _________

________ groups of _________

________ groups of _________

Product

Options for 12 girls =

2 x−1

x+2

14

?

? 20 x

?

?

EX 4: What are the side lengths/factors if the area (product) is x2+ x?

Arrange algebra tiles for x2 and x into a rectangle.Draw and label the squares and rectangles on your paper.

Label the side lengths to show what factors would give you the product in each rectangle.

EX 5: Represent 2 x2+5 x+3 as a rectangle with algebra tiles. Draw and label the squares and rectangles on your paper.

What are the factors/side length that can be multiplied to make 2 x2+5 x+3? Prove it symbolically.

EX 6: Find the factors of x2−6 x+8. Flip the Algebra tiles over to reflect negative values. Draw the rectangle with algebra tiles on your notes.

Try these next two with your table partner. Find the factors.

7. 2 x2+9 x+10 8. x2−5 x+6

x2+ x

?

?

FACTORING INTRO – PART 2 NAME: __________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

In previous lessons, you were asked to find the key features of quadratic functions. What were the key features and how did you find them?

Line/Axis of Symmetry Vertex y-intercept

Domain Range x-intercept(s)

A. The following functions are in factored form (not standard form). Label the x-intercepts in the graphs and discuss the relationship between the x-intercepts and the equations.

f ( x )=−( x+2 ) ( x−4 ) f ( x )=0.22 (2 x−1 ) (2 x−7 ) f ( x )=x ( x+5 )

Describe the relationship between the x-intercepts and the factors.

B. Label the algebra tiles for the product x2+ x.

Label the side lengths to show what factors would give you the product in each rectangle.

Write the sides as a product: _________ ________________

How does the graph of f ( x )=x2+x confirm the product?

( 72,0)( 1

2,0)

.6

(−2 ,0 ) (4 ,0 )

(−5 ,0 ) (0 ,0 )

Use algebra tiles to write the factors of the function. Write the function as a product of factors. Substitute either one of the x-coordinates of the x-intercepts into the factors and evaluate the product.

C. f ( x )=2x2+5x+3

f ( x )=¿ ( )( )

D. f ( x )=x2−6x+8

f ( x )=¿ ( )( )

.6.6.6

Use algebra tiles to write the factors of the function. Write the function as a product of factors. Substitute either one of the x-coordinates of the x-intercepts into the factors and evaluate the product.

E. f ( x )=2x2−9 x+10

f ( x )=¿ ( )( )

F. f ( x )=x2+5 x

f ( x )=¿ ( )( )

SUMMARY: What happens when you substitute the x-coordinate into the equation? Why is this ALWAYS true?

.6 .6.6

HOW TO FACTOR: WHAT IS A GCF? NAME: _________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

Focus Question: What is a GCF?

What if our trinomial is 10 x2+60 x+50 and we need to find the factors? Why is this a problem? What can we do to fix this problem?

What is a GCF? The greatest _______________________ ___________________that divides into each term.

What is the GCF of 27 and 36? Of 16a2and 24 a5?

When you “factor out the GCF” what happens? EX A: 24 p4– 6 p2+3 p EX B: a2+a EX C: 10 x2+60 x+50

EX D: −12 x2+4 x+8

EX E: −x2−x+6

How is example D different from the other examples?

If the first term (when written in _________________________ order)

is negative, then you need to divide out a ________________ GCF!

HOW TO FACTOR: WHAT IS THE ZPP? NAME: _________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

Focus Question: What is the ZPP and why would you need it?

Watch the process and then describe it below.

A. 0=10 x2+50 x+10x+50

Write equation in factored form:

0=¿

Use the ZPP.

Graph theintercepts and sketch the parabola.

B. 0=2x2−5 x−4 x+10

Write equation in factored form:

0=¿

Use the ZPP.

Graph theintercepts and sketch the parabola.

Process:

1. Check for a __________ (especially a negative one,) and “reverse distribute” if you need to.

2. Write each ___________ in a portion of the area model.

3. Factor out a _________ from each row, columns are labeled with the other _____________.

4. Write the result as the ____________________ of two _______________.

5. Set each factor equal to __________________and ___________________ for x. This is the ZPP!

C. 0=−12x2+12 x−8 x+8

Write equation in factored form:

0=¿

Use the ZPP.

Graph theintercepts and sketch the parabola.

D. 0=x2 +x+2 x+2

Write equation in factored form:

0=¿

Use the ZPP.

Graph theintercepts and sketchthe parabola.

Summary:What does ZPP stand for?

How do you use it? Why would you use it?

HOW TO FACTOR: TRINOMIALS DAY #1 NAME: _________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

Focus Question: Could you go backwards and “take apart” a trinomial?

For example 2 x3+11 x2+5 x = ( ) ( ) ?????

1. The first thing you do is check for a __________ (especially a negative one,) and “reverse distribute” if you need to.

2 x3+11 x2+5 x =

2. Set up an area model. Write in the first and last terms.

3. Then ask yourself, “What multiplied by what

will make the _________ __________?

4. Make a list of __________ ___________ for the last term.

5. Trial and error with the pairs and the signs until you find the right one.

6. Write your answer as a product of two binomials.

−x2+3x+28 x2−7 x+2

Factors: Factors:

last

first

Try two more AND complete the process to find the x-intercepts and graph the parabola

1. GCF?

2. “What multiplied by what will make the ___________ _________?

3. List of factor pairs for the ____________ ____________.

4. Trial and error with the pairs and the signs until you find the right one. Make sure you multiply the binomials together to check your work.

5. 0 = GCF( )( )

6. ZPP

7. Graph the intercepts and sketch the parabola.

y=¿ 2 x2 – 9x+10

0=¿

Use the ZPP.

Graph theintercepts and the parabola.

y=¿ −12 x2+4 x+8

0=¿

Use the ZPP.

Graph theintercepts and the parabola.

HOW TO FACTOR: TRINOMIALS DAY #2 NAME: _________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

Focus Question: How do you factor expressions when there are more choices?

1. The first thing you do is check for a __________ (especially a negative one,) and “reverse distribute” if you need to.

2. Set up an area model. Write in the first and last terms.

3. Then ask yourself, “What multiplied by what

will make the _________ __________?

4. Make a list of factor pairs for the __________ ___________.

5. Trial and error with the pairs and the signs until you find the right one.

6. Write your answer as a product of two binomials.

3 x2– 20x+12 −48x2+4 x+2

Factors: Factors:

last

first

What about without the area model?

1. GCF?

2. 0 = GCF( )( )

3. “What multiplied by what will make the ___________ _________?

4. List of factor pairs for the ____________ ____________.

5. Trial and error with the pairs and the signs until you find the right one. Make sure you multiply the binomials together to check your work.

6. ZPP

f (a)=−4a3−44 a2−96a f (x)=9 x2−6 x+1

HOW TO FACTOR: ONLY TWO TERMS NAME: _________________________________A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

Focus Question: What do you do if there are only two terms to factor?

What is the difference between the two columns?

Factor: x2+3x

2a2−100a

−9 f 2−56 f

x2−x

−16 t 2+80t

x2−25

2a2−200

−9 f 2+49 g2

7 x2– 7 y2

36−49 p2

What do you notice?

Find the x-intercepts and sketch the parabola.1. GCF?2. Factor.3. ZPP4. Sketch the graph of the parabola.

y=x2+25 x f (x)=x2−25

f (a)=2a2−200a y=2a2−200