Theory of Intelligence
9TH•Mathematics•Factorization•4.1.3 Types II–V Factorization

Type – II: Factorization of the expression of the types $a^4 + a^2b^2 + b^4$ or $a^4 + b^4$

Factorize a⁴+b⁴, grouped products, perfect cubes, and sum/difference of cubes.

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Type – II: Factorization of the expression of the types a4+a2b2+b4a^4 + a^2b^2 + b^4 or a4+b4a^4 + b^4

Let's factorize the first expression

a4+a2b2+b4a^4 + a^2b^2 + b^4 =a4+b4+a2b2= a^4 + b^4 + a^2b^2 =(a2)2+(b2)2+a2b2= (a^2)^2 + (b^2)^2 + a^2b^2 =(a2)2+(b2)2+2a2b2−2a2b2+a2b2(Adding and subtracting 2a2b2)= (a^2)^2 + (b^2)^2 + 2a^2b^2 - 2a^2b^2 + a^2b^2 \quad \text{(Adding and subtracting } 2a^2b^2\text{)} =(a2+b2)2−a2b2= (a^2 + b^2)^2 - a^2b^2 =(a2+b2)2−(ab)2= (a^2 + b^2)^2 - (ab)^2 =(a2+b2−ab)(a2+b2+ab)= (a^2 + b^2 - ab)(a^2 + b^2 + ab) =(a2−ab+b2)(a2+ab+b2)= (a^2 - ab + b^2)(a^2 + ab + b^2)
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Remember!
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Example 9

Factorize: x4+x2+25x^4 + x^2 + 25

Solution: x4+x2+25x^4 + x^2 + 25

=x4+25+x2= x^4 + 25 + x^2 =(x2)2+(5)2+2(x2)(5)−2(x2)(5)+x2(Adding and subtracting 2(x2)(5))= (x^2)^2 + (5)^2 + 2(x^2)(5) - 2(x^2)(5) + x^2 \quad \text{(Adding and subtracting } 2(x^2)(5)\text{)} =(x2+5)2−10x2+x2= (x^2 + 5)^2 - 10x^2 + x^2 =(x2+5)2−9x2= (x^2 + 5)^2 - 9x^2 =(x2+5)2−(3x)2= (x^2 + 5)^2 - (3x)^2 =(x2+5−3x)(x2+5+3x)= (x^2 + 5 - 3x)(x^2 + 5 + 3x) =(x2−3x+5)(x2+3x+5)= (x^2 - 3x + 5)(x^2 + 3x + 5)
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Activity

  • Prepare cards by writing several expressions.
  • Divide students in small groups.
  • Each group will draw a card and factorize the expression.
  • The group which completes the most correct factorizations in a set time will win.

Example 10

Factorize: x4+y4x^4 + y^4

Solution: x4+y4x^4 + y^4

=(x2)2+(y2)2= (x^2)^2 + (y^2)^2 =(x2)2+(y2)2+2(x2)(y2)−2(x2)(y2)(Adding and subtracting 2x2y2)= (x^2)^2 + (y^2)^2 + 2(x^2)(y^2) - 2(x^2)(y^2) \quad \text{(Adding and subtracting } 2x^2y^2\text{)} =(x2+y2)2−(2xy)2= (x^2 + y^2)^2 - (\sqrt{2}xy)^2 =(x2+y2−2xy)(x2+y2+2xy)= (x^2 + y^2 - \sqrt{2}xy)(x^2 + y^2 + \sqrt{2}xy) =(x2−2xy+y2)(x2+2xy+y2)= (x^2 - \sqrt{2}xy + y^2)(x^2 + \sqrt{2}xy + y^2)
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Try Yourself!
Factorize: (i) 64x4y4+z464x^4y^4 + z^4
(ii) 81x4+181x4−1181x^4 + \dfrac{1}{81x^4} - 11

Example 11

Factorize: a4+64a^4 + 64

Solution: a4+64a^4 + 64

=(a2)2+(8)2= (a^2)^2 + (8)^2 =(a2)2+(8)2+2(a2)(8)−2(a2)(8)(Adding and subtracting 2(a2)(8))= (a^2)^2 + (8)^2 + 2(a^2)(8) - 2(a^2)(8) \quad \text{(Adding and subtracting } 2(a^2)(8)\text{)} =(a2+8)2−16a2= (a^2 + 8)^2 - 16a^2 =(a2+8)2−(4a)2= (a^2 + 8)^2 - (4a)^2 =(a2+8−4a)(a2+8+4a)= (a^2 + 8 - 4a)(a^2 + 8 + 4a) =(a2−4a+8)(a2+4a+8)= (a^2 - 4a + 8)(a^2 + 4a + 8)

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