# Portfolio of Problems on Special Products and Factoring

Submitted by: Alaina Palomo

Onyx

Submitted to: Sir Eslem Tonido

Special Products

A. Multiplication of Polynomials by Monomials

4a2 (3a + 5)

=(4a2 ∙ 3a) = (4a2 + 5)

= 12a3 + 20a2

B. Multiplication of 2 binomials

(4x – 4);(2x + 5)

4x – 4

2x + 5

8x2 + 8x

+20x – 20

8x2 + 12x – 20 Final Product

C. Square of binomials ( Perfect Square Trinomials)

(2x + 3)(2x + 3)

Using Foil Method:

=(2x)(2x) + (2x)(3) + (3)(2x) + 3(3)

=4x2 + 6x + 6x + 9

=4x2 + 12x + 9

= 4x2 + 12x + 9 Final Product

D. Product of the sum and different two terms

1.) (x+6)(x+6) = x2 – 36

2.) (½ x – 1) (½ x + 1 ) = ¼ x2 – 1

E. Cube of Binomial

A. (x + y)3= x3 + 3x2y + 3xy2 + y3

B. (x – y)3 = x3 - 3x2y + 3xy2 - y3

1.) (2x + 3y)3 = x3 + 3x2y + 3xy2 + y3

X = 2x =(2x)3 + 3(2x)2(3y) + 3(2x)(3y)2 + (3y)3

Y = 3y = 8x3 + 36x2y + 54xy2 + 27y3

2.) (5x – 3)3 = x3 - 3x2y + 3xy2 - y3

X = 5x = (5x)3 – 3(5x)2(3) + 3(5x)(3)2 – (3)3

Y = 3 = 125x3 – 225x2 + 135x – 27

F. Square of Trinomial

( x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz

1.) ( 2x + 3y – 4)2 = (2x)2 + (3y)2 + (-4)+2 + 2(2x)(3x) + 2(2x)(-4) X= 2x + 2(3y)(-4)

Y = 3y = 4x+2 + 9y2 + 16 + 12xy – 16x – 24y Z = -4

G. Sum and difference of two cubes

1.) (a + b) (a2 – ab + b2) = a3 + b3

2.) (3x + 2) (9x2 – 6x + 4) = 27x3 + 8

3.) (a – b) (a2 + ab + b2) = a – b3

4.) (5a- 4b) (35a2 + 20ab + 16b2 = 125a3 – 64b3

5.) (3x + 2y) (9x2 – 6xy + 4y2) = 27x3 + 8y3

Factoring

A. Common Monomial Factor

Procedures:

a) Find the GCF of the terms in the polynomial.

b) divide each term by the GCF to get the other factors

1.) 9x+4 + 27x2y – 63x5y2

9 = 3 ∙ 3

27 = 3 ∙ 3 ∙ 3

63 = 3 ∙ 3 ∙ 7

3 ∙ 3

GCF = 9

= 9x2 (x2 + 3y – 7xy2)

B. Factoring the difference of two squares

1.) x2 – y2 = (x + y)(x - y)

2.) 9a2 – 25b2 = (3a – 5b)(3a + 5b)

3.)a8x4 – 16 = (a4 x2 +4)(a4x2 -4

4.)(2x-y)2 – 25z2 = [(2x – y) – 5z][(2x –y) + 5z]

5.) y4 – 16 = (y2 - 4)(y2 + 4)

(y+2)(y-2)(y2 + 4)

C. Factoring Perfect Square Trinomial

Procedure:

a) Get the square roots of the first and last terms.

b) use the sing in the middle term between these roots.

c) square the binomial obtained on step b.

1.) x2 + 14x + 9

(x + 3)2

2.) 1 – 12x + 36x2

(1 – 6x)2

3.) 49x2 – 5xy + 16y2

(7 – 4y)2

4) 64 + 48xy + 9 x2y2

(8 + 9xy)2

5.) x2+ + 16x + 64

(x + 8)2

D. Finding the missing term

* Finding the Last Term

1.) x2 - 8x + 16

X 2

4x = (4)2

X 16

2.) 4x2 + 20x + 25

2x 2

10 = (5)2

2x 25

* Finding the First Term

1.) 64 + 48xy = 9x2y2

2 3xy

24xy = (8)2

3xy

* Finding the Middle Term

1.) 16x2 + 8x + 1

4x 1

2(4x)

= 8x

2.) 25x2 + 30x + 3

5x 3

2(15x)

= 30x

E. Factoring Quadratic Trinomial of the term x2 + bx + c

1.) a2 + 9a + 20

(a + 4)(a + 5)

+20

2 ∙ 10

-2 ∙ -10

1 ∙ 20

-1 ∙ 20

5 ∙ 4

-5 ∙ -4

2.) a2 + 3a – 10

( a + 5 )( a – 2)

-10

-5 ∙ 2

5 ∙ 2

10 ∙ -1

-10 ∙ 1

F. Factoring the sum or difference of cubes

1.) 64x6 - y9 = (4x2 - y3)(16x4 + 4x2y3 + y3)

2.) 8x3 + 27 = (2x + 3) (4x2 – 6x + 9)

3.) 125a6 – 512b3 = )5a2 – 8b)(25a4 + 40a2b = 64b2)

Please join StudyMode to read the full document