Kakaloka yung factoring exercises! Nakakasabaw ng utak! swear! Anyway, Sinusulat ko tong blog post na ito para sa dalawang dalagitang nagpaturo ng Algebra saken kanina na saksakan ng HIRAP!!!! Wahahah! Hindi ako makakatulog pag hindi ko naexplain ng maayos yung topic kya eto ang follow up discussion! Sana matutunan nyo yung process!

#1/p45. 16x² - 8xy + y² + 12x - 3y + 2

For this type of expressions, eto yung mga sagot sa Product of two Trinomials. Makikita nyo kagad yun pag may tatlong components: may x and y in 2nd power; May xy term; at may linear x and y plus constant.

Let us review yung actual Product of two Trinomials:

(A₁ + B₁ + C₁)(A₂ + B₂ + C₂) = ...

A₁A₂ + A₁B₂ + A₁C₂ + B₁A₂ + B₁B₂ + B₁C₂ + C₁A₂ + C₁B₂ + C₁C₂

Now in the sample expression, yung x and y in the 2nd power terms: 16x² tska y², ay parehong perfect square. Therefore we can assume na yung both 1st and 2nd terms (A and B) are the same for the two Trinomials. Meaning:

A₁ = A₂ = A = 4x --> square root ng 16²
B₁ = B₂ = B = +/- y --> square root ng y²

Remember, this is just an assumption. Kasi pede yung 16x² eh product ng 8x at 2x. Pero magandang assumption na din yun para may simulan ka. One thing that you wil llearn in engineering eh lagi kayong mag-aassume.

Take note dun sa square root ng y² may +/- signs sya, Pero dun sa 16x² wala. Gawin nyo din assumption na yung x term laging positive pag kinukuha yung square root. Macocompesate naman yung sign dun sa ibang factors.

So palitan na natin lahat nag A₁ at A₂ to the gerneral term A. Tapos yung mga B₁ at B₂ to the general term B.

Yung product ay magigig ganito na:

= AA + AB + AC₂ + AB + BB + BC₂ + AC₁ + BC₁ + C₁C₂

SIMPLIFYING:
= A² + 2AB + AC₂ + B² + BC₂ + AC₁ + BC₁ + C₁C₂

SIMPLIFYING FURTHER FACTORING OUT C₁ AND C₂
= A² + 2AB + (A + B)C₂ + B² + (A + B)C₁ + C₁C₂

MORE FURTHER SIMPLICIFACTION: FACTORING (A + B)
= A² + 2AB + (A + B)(C₁+C₂) + B² + C₁C₂

REARRAGING:
= A² + 2AB + B² + (A + B)(C₁+C₂) + C₁C₂

LOOKS FAMILIAR???

Factor natin yung 12x - 3y magiging (4x - y)(3). Ano napapansin nyo sa (4x - y) hindi ba yun yung (A + B) natin? Therefore yung 3 ay C₁+C₂. Tapos yung square root ng y² ay -y pala dahil sa lumabas after factoring out 3.

Masosolve natin yung C₁+C₂ ng ganito:

C₁+C₂ = 3 -> eq 1
C₁C₂ = 2 -> eq 2

you have two equations with two unknowns, masosolve nyo yung dalawa respectively.

Pero para mas madali, logic na lang gamitin natin. Anung factors ng 2 na ang sum ay 3? edi 2 and 1!

Edi ang final answer pala ay...

(4x - y + 2) (4x - y +1)

ANG HABA!!!


Solve naman natin ngayon using the improved WILBERCHIE Method!!!

16x² - 8xy + y² + 12x - 3y + 2
Dahil perfect square ang x and y in 2nd power, assume mong preho yung 1st term at positive sya, assume mo rin na pareho yung 2nd term

A = 4x
B = +/- y

Divide natin yung A sa xy term

^-8xy/_4x = -2y --> 2 times B, special property pag pareho ang 1st and 2nd terms.

^-2y/₂ = -y --> B

Divide mo naman ngayon yung nasolve mong B Dun sa y term;

^-3y/_-y = 3 --> sum of the constants.

or yung A sa x term

^12x/_4x = 3 --> sum of the constants.

Product of constants yung given constant term sa expression = 2.

By logic, Factors of 2 whose sum is 3. Answer: 2 and 1.

Final Answer: (4x - y + 2) (4x - y +1)

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#2/p45. x⁸ +64

Completing the Square: ibigay yung nawawalang constant Mid term = 2*sqrt(1st and last terms)
=x⁸ + 16x⁴ + 64 - 16x⁴
=(x⁸ + 16x⁴ + 64) - 16x⁴

factor using PST:
=(x⁴ + 8)² - 16x⁴

Factor using Difference of Two squares
=(x⁴ + 8 + 4x²)(x⁴ + 8 - 4x²) <-- Final Answer