Factoring, Difference of Two Squares

04/20/23

Polynomial Factoring

Objective: factor a quadratic expression where the a term does not equal one.

Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

Notes• Magic X

8

64 2

Ex.

10

75 2

Ex.

35

27 5

Ex.

10

35 2

Ex.

24

1012 2

Ex.

6

76 1

Ex.

Notes• Steps for Factoring a Quadratic Equation

1) Factor out the GCF (if possible)

2) Set up the Magic X.

• Find the pair that multiplies to equal the top, and adds up to equal the bottom.

3) Set up the Magic T.• Put ax on the top two, and the pair we found for the bottom two.• Treat each side as a fraction and reduce if possible

4) You now have your factors.

2ax bx c

a c

b1# 2#

ax ax

1# 2#

• ac goes on top, and b goes on bottom

Notes

1) Factor out GCF.

30

Factor. Ex.

23 11 10x x

115 6

3x 3x

5 6

2) Magic X.

3) Magic T.

- Reduce?

- What pair multiplies to equal top and adds to equal the bottom?

1

24) You now have your factors.

5 6 30

5 6 11 3 10 30

3 5x 2x 3 5 2x x GCF 1

Notes

1) Factor out GCF.

24

Factor. Ex.

22 5 12x x

53 8

2x 2x

3 8

2) Magic X.

3) Magic T.

- Reduce?

- What pair multiplies to equal top and adds to equal the bottom?

1

44) You now have your factors.

3 8 24

3 8 5 2 12 24

2 3x 4x 2 3 4x x

GCF 1

Notes

1) Factor out GCF.

20

Factor. Ex.

22 9 10x x

94 5

2x 2x

4 5

2) Magic X.

3) Magic T.

- Reduce?

- What pair multiplies to equal top and adds to equal the bottom?

1

2 4) You now have your factors.

4 5 20

4 5 9 2 10 20

2x 2 5x 2 2 5x x

Now you try.

GCF 1

Notes

1) Factor out GCF.

24

Factor. Ex.

26 5 4x x

53 8

6x 6x

3 8

2) Magic X.

3) Magic T.

- Reduce?

- What pair multiplies to equal top and adds to equal the bottom?

2

14) You now have your factors.

3 8 24

3 8 5 6 4 24

2 1x 3 4x 2 1 3 4x x

3

4

GCF 1

Notes

1) Factor out GCF.

20

Factor. Ex.

210 2x x

14 5

10x 10x

4 5

2) Magic X.

3) Magic T.

- Reduce?

- What pair multiplies to equal top and adds to equal the bottom?

5

24) You now have your factors.

4 5 20

4 5 1 10 2 20

5 2x 2 1x 5 2 2 1x x

2

1

GCF 1

Notes

1) Factor out GCF.

48

Factor. Ex.

216 16 3x x

164 12

16x 16x

4 12

2) Magic X.

3) Magic T.

- Reduce?

- What pair multiplies to equal top and adds to equal the bottom?

4

1 4) You now have your factors.

4 12 48

4 12 16 16 3 48

4 1x 4 3x 4 1 4 3x x

4

3

Now you try.

GCF 1

Notes• Difference of Two Squares

2. Both terms must be perfect squares.

2 2a b a b a b

1. Must be subtraction.

Determine if it the problem is a difference of two squares. Ex.

29 16x Yes

Ex.2 64x Yes

Ex.24 10x No

Ex.2 100x Yes

Ex.249 20x No

Ex.2 16x xNo

Now you try.

NotesFactor.

Ex.2 36x

x x 6 6x x6 6 6 6x x

Ex.281 49x

9x 9x 7 79x 9x7 7 9 7 9 7x x

Ex.216 25x

216 25x

Now you try.

Ex.2 64x

x x 8 8x x8 8 8 8x x

Ex.225 36x

5x 5x 6 6

5x 5x6 6 5 6 5 6x x

Ex.2 9x x

x x 9 9x x

GCF x

Summary To factor a quadratic first set up the Magic __.

On top multiply ac and on _______ put b. Now find a pair of numbers that ________ to equal the top and ____ to equal the bottom. Next set up your ______ T. Put ax on the top and put your pair on the bottom. Reduce if possible.

Xbottom

multiplyadd Magic

You have a difference of ____ squares if both terms are _______ squares and it is a __________ expression. If so, just break up each term and make one parenthesis ________ and the other subtraction.

twosubtractionperfect

addition

Class Work / Homework

1. Pencil ONLY.2. Must show all of your work.• NO WORK = NO CREDIT

3. Must attempt EVERY problem.4. Always check your answers.

Rules for Homework

Class Work / HomeworkFactor.

21. 12 36x x 22. 18 81x x 23. 14 49x x

24. 4 12 9x x 25. 9 12 4x x 26. 6 7 2x x

27. 6 5 6x x 28. 8 6 9x x

29. 12 13 4x x 210. 10 7 12x x