1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§5.5 Factor Special Forms
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 5.4 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§5.4 → Factoring TriNomials
Any QUESTIONS About HomeWork
§5.4 → HW-14

3. §5.5 Factoring Special Forms
Factoring Perfect-Square Trinomials and Differences of Squares
Recognizing Perfect-Square Trinomials
Factoring Perfect-Square Trinomials
Recognizing Differences of Squares
Factoring Differences of Squares
Factoring SUM of Two Cubes
Factoring DIFFERENCE of Two Cubes

4. Recognizing Perfect-Sq Trinoms
A trinomial that is the square of a binomial is called a perfect-square trinomial
A2 + 2AB + B2 = (A + B)2
A2 − 2AB + B2 = (A − B)2
Reading the right sides first, we see that these equations can be used to factor perfect-square trinomials.
A2 + 2AB + B2 = (A + B)(A + B)
A2 − 2AB + B2 = (A − B)(A − B)

5. Recognizing Perfect-Sq Trinoms
Note that in order for the trinomial to be the square of a binomial, it must have the following:
1. Two terms, A2 and B2, must be squares, such as: 9, x2, 100y2, 25w2
2. Neither A2 or B2 is being SUBTRACTED.
3. The remaining term is either 2  A  B or −2  A  B
where A & B are the square roots of A2 & B2

6. Example  Trinom Sqs
Determine whether each of the following is a perfect-square trinomial.
a) x2 + 8x b) t2 − 9t − 36
c) 25x2 + 4 – 20x
SOLUTION a) x2 + 8x + 16
Two terms, x2 and 16, are squares.
Neither x2 or 16 is being subtracted.
The remaining term, 8x, is 2x4, where x and 4 are the square roots of x2 and 16

7. Example  Trinom Sqs SOLUTION b) t2 – 9t – 36
Two terms, t2 and 36, are squares.
But 36 is being subtracted so t2 – 9t – 36 is not a perfect-square trinomial
SOLUTION c) 25x2 + 4 – 20x
It helps to write it in descending order.
25x2 – 20x + 4

8. Example  Trinom Sqs SOLUTION c) 25x2 − 20x + 4
Two terms, 25x2 and 4, are squares.
There is no minus sign before 25x2 or 4.
Twice the product of the square roots is 2  5x  2, is 20x, the opposite of the remaining term, −20x
Thus 25x2 − 20x + 4 is a perfect-square trinomial.

9. Factoring a Perfect-Square Trinomial
The Two Types of Perfect-Squares
A2 + 2AB + B2 = (A + B)2
A2 − 2AB + B2 = (A − B)2

10. Example  Factor Perf. Sqs
Factor: a) x2 + 8x + 16
b) 25x2 − 20x + 4
SOLUTION a)
x2 + 8x + 16 = x2 + 2  x  = (x + 4)2
A A B + B2 = (A + B)2

11. Example  Factor Perf. Sqs
Factor: a) x2 + 8x + 16
b) 25x2 − 20x + 4
SOLUTION b)
25x2 – 20x + 4 = (5x)2 – 2  5x  = (5x – 2)2
A2 – 2 A B + B2 = (A – B)2

12. Example  Factor 16a2 – 24ab + 9b2
SOLUTION
16a2 − 24ab + 9b2 = (4a)2 − 2(4a)(3b) + (3b)2
= (4a − 3b)2 = (4a − 3b)(4a − 3b)
CHECK:
(4a − 3b)(4a − 3b) = 16a2 − 24ab + 9b2 
The factorization is (4a − 3b)2.

13. Expl  Factor 12a3 – 108a2 + 243a SOLUTION
Always look for a common factor. This time there is one. Factor out 3a.
12a3 − 108a a = 3a(4a2 − 36a + 81)
= 3a[(2a)2 − 2(2a)(9) + 92]
= 3a(2a − 9)2
The factorization is 3a(2a − 9)2

14. Recognizing Differences of Squares
An expression, like 25x2 − 36, that can be written in the form A2 − B2 is called a difference of squares.
Note that for a binomial to be a difference of squares, it must have the following.
There must be two expressions, both squares, such as: 9, x2, 100y2, 36y8
The terms in the binomial must have different signs.

15. Difference of 2-Squares
Diff of 2 Sqs → A2 − B2
Note that in order for a term to be a square, its coefficient must be a perfect square and the power(s) of the variable(s) must be even.
For Example 25x4 − 36
25 = 52
The Power on x is even at 4 → x4 = (x2)2
Also, in this case 36 = 62

16. Example  Test Diff of 2Sqs
Determine whether each of the following is a difference of squares.
a) 16x2 − 25 b) 36 − y c) −x
SOLUTION a) 16x2 − 25
The 1st expression is a sq: 16x2 = (4x)2
The 2nd expression is a sq: 25 = 52
The terms have different signs.
Thus, 16x2 − 25 is a difference of squares, (4x)2 − 52

17. Example  Test Diff of 2Sqs
SOLUTION b) 36 − y5
The expression y5 is not a square.
Thus, 36 − y5 is not a diff of squares
SOLUTION c) −x
The expressions x12 and 49 are squares: x12 = (x6)2 and 49 = 72
The terms have different signs.
Thus, −x is a diff of sqs, 72 − (x6)2

18. Factoring Diff of 2 Squares
A2 − B2 = (A + B)(A − B)
The Gray Area by Square Subtraction
The Gray Area by (LENGTH)(WIDTH)

19. Example  Factor Diff of Sqs
Factor: a) x2 − 9 b) y2 − 16w2
SOLUTION a) x2 − 9 = x2 – 32 = (x + 3)(x − 3)
A2 − B2 = (A + B)(A − B)
b) y2 − 16w2 = y2 − (4w)2 = (y + 4w)(y − 4w)
A2 − B2 = (A + B) (A − B)

20. Example  Factor Diff of Sqs
Factor: c) 25 − 36a12 d) 98x2 − 8x8
SOLUTION
c) 25 − 36a12 = 52 − (6a6)2 = (5 + 6a6)(5 − 6a6)
d) 98x2 − 8x8
Always look for a common factor. This time there is one, 2x2:
98x2 − 8x8 = 2x2(49 − 4x6)
= 2x2[(72 − (2x3)2]
= 2x2(7 + 2x3)(7 − 2x3)

21. Grouping to Expose Diff of Sqs
Sometimes a Clever Grouping will reveal a Perfect-Sq TriNomial next to another Squared Term
Example Factor m2 − 4b4 + 14m + 49
 rearranging 
m2 + 14m + 49 − 4b4
 GROUPING 
(m2 + 14m + 49) − 4b4

22. Grouping to Expose Diff of Sqs
Example Factor m2 − 4b4 + 14m + 49
Recognize m2 + 14m + 49 as Perfect Square Trinomial → (m+7)2
Also Recognize 4b4 as a Sq → (2b)2
(m2 + 14m + 49) − 4b4
 Perfect Sqs 
(m + 7)2 − (2b2)2
In Diff-of-Sqs Formula: A→m+7; B→2b2

23. Grouping to Expose Diff of Sqs
Example Factor m2 − 4b4 + 14m + 49
(m + 7)2 − (2b2)2
 Diff-of-Sqs → (A − B)(A + B) 
([m+7] − 2b2)([m + 7] + 2b2)
 Simplify → ReArrange 
(−2b2 + m + 7)(2b2 + m + 7)
The Check is Left for us to do Later

24. Factoring Two Cubes
The principle of patterns applies to the sum and difference of two CUBES. Those patterns
SUM of Cubes
DIFFERENCE of Cubes

25. TwoCubes SIGN Significance
Carefully note the Sum/Diff of Two-Cubes Sign Pattern
SAME Sign
OPP Sign
SAME Sign
OPP Sign

26. Example: Factor x3 + 64 Factor Recognize Pattern as Sum of CUBES
Determine Values that were CUBED
Map Values to Formula
Substitute into Formula
Simplify and CleanUp

27. Example: Factor 8w3−27z3 Factor
Recognize Pattern as Difference of CUBES
Determine CUBED Values
Simplify by Properties of Exponents
Map Values to Formula
Sub into Formula
Simplify & CleanUp

28.  Example: Check 8w3−27z3 Check Use Distributive property
Use Comm & Assoc. properties, and Adding-to-Zero 

29. Sum & Difference Summary
Difference of Two SQUARES
SUM of Two CUBES
Difference of Two CUBES

30. Factoring Completely
Sometimes, a complete factorization requires two or more steps. Factoring is complete when no factor can be factored further.
Example: Factor 5x4 − 3125
May have the Difference-of-2sqs TWICE

31. Factoring Completely SOLUTION 5x4 − 3125 = 5(x4 − 625)
= 5(x − 5)(x + 5)(x2 + 25)
The factorization: 5(x − 5)(x + 5)(x2 + 25)

32. Factoring Tips
Always look first for a common factor. If there is one, factor it out.
Be alert for perfect-square trinomials and for binomials that are differences of squares.
Once recognized, they can be factored without trial and error.
Always factor completely.
Check by multiplying.

33. WhiteBoard Work Problems From §5.5 Exercise Set
14, 22, 48, 74, 94, 110
The SUM (Σ) & DIFFERENCE (Δ) of Two Cubes

34. All Done for Today
Sum of Two Cubes

35. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

36. Graph y = |x|
Make T-table