1. Sum and Difference of Two Terms
Math 2
Special Products:
Sum and Difference of Two Terms
Math San Beda College Mr. Jay Mar Bolajo

2. Use FOIL to multiply:
(x – 5) (x + 5)
(x + 7) (x - 7)
(3a – b) (3a + b)
x2 - 25
x2 - 49
a2 – b2
(y + 11) (y - 11)
(2n – 5) (2n + 5)
(x – 3) (x + 3) y
n2 - 1
x2 - 9
What is the pattern you observed?

3. Product of a Sum and Difference
“The product of (a + b) and (a – b) is
the square of a minus the square of b.”
(a + b) (a – b)
= a2 – ab + ab – b2
The middle terms cancel because they’re opposites.
= a2 – b2
The result is the difference of the squares of the two original terms.

4. Multiply. Use the shortcut.
(3x + 8y) (3x – 8y)
Shortcut:
Square the first term and
subtract the square of the second term.
= (3x)² - (8y)2
= 9x² - 64y2

5. Try these! (x + 7) (x – 7) x²- 49 16t²- 1 (4t + 1)(4t – 1)
(9x – 5y)(9x + 5y)
81x²- 25y²
(-3x + 5)(-3x – 5)
9x²- 25

6. Multiply the binomials using FOIL:
(3n + 7) (3n – 7)
(2x - 5) (2x + 5)
= 9n2 – 21n + 21n – 49
= 9n2 - 49
= 4x2 + 10x – 10x – 25
= 4x2 - 25
(8 + a) (8 – a)
(x + 6y) (x – 6y)
= 64 – 8a + 8a – a2
= 64 – a2
= x2 – 6xy + 6xy – 36y2
= x2 – 36y2
The result is the square of the first term minus the square of the second term.

7. Use FOIL to multiply:
(x + 6) (x + 6)
(x + 9) (x + 9)
(a + b) (a + b)
x2 + 12x + 36
x2 + 18x + 81
a2 + 2ab + b2
(y + 10) (y + 10)
(n + 1) (n + 1)
(x + 3) (x + 3)
y2 + 20y + 100
n2 + 2n + 1
x2 + 6x + 9
What is the pattern you observed?

8. Square of a Sum “The square of (a + b) is
the square of a plus twice the product of a and b plus the square of b.”
(a + b) (a + b)
= a2 + ab + ab + b2
The middle terms are identical.
= a2 + 2ab + b2
The result is the square of a, plus two times the product of a and b, plus the square of b.

9. Use FOIL to multiply:
(x - 2) (x - 2)
(x - 17) (x - 17)
(a - b) (a - b)
x2 - 4x + 4
x2 - 34x + 289
a2 - 2ab + b2
(y - 10) (y - 10)
(n - 9) (n - 9)
(x - 3) (x - 3)
y2 - 20y + 100
n2 - 18n + 81
x2 - 6x + 9
What is the pattern you observed?

10. Square of a Difference “The square of (a – b) is
the square of a minus twice the product of a and b plus the square of b.”
(a - b) (a - b)
= a2 - ab - ab + b2
The middle terms are identical.
= a2 - 2ab + b2
The result is the square of a, minus two times the product of a and b, plus the square of b.

11. The Square of a Binomial Pattern
(x + 6)2
The binomial (x + 6) is being squared.
= (x + 6) (x + 6)
Expand before multiplying.
= x2 + 6x + 6x + 36
Even though you may remember the pattern, you still need to use FOIL. This will help you INTERNALIZE the patterns so you’ll recognize them when we begin factoring.
= x2 + 12x + 36

12. (3n - 5)2
(7z + 2)2
= (3n - 5) (3n - 5)
= (7z + 2) (7z + 2)
= 9n2 – 15n – 15n + 25
= 49z2 + 14z + 14z + 4
= 9n2 -30n + 25
= 49z2 + 28z + 4

13. Some good advice……… Don’t be fooled! (a + b)2 DOES NOT mean a2 + b2
(a + b)2 means (a + b) (a + b)
Use FOIL on this assignment. Yes, you should recognize the patterns that are occurring, but don’t rely on your memory of them. Use FOIL to help your brain internalize the pattern. The recognition of the pattern will be most useful during factoring