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The document outlines various methods of factorization in mathematics, including examples and solutions for quadratic expressions, perfect squares, and expansion formulas. It emphasizes the importance of understanding mathematics beyond just calculations. The author has over 8 years of experience and has mentored thousands of students, including top JEE rank holders.

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Ishan 8th B
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7 views24 pages

Original

The document outlines various methods of factorization in mathematics, including examples and solutions for quadratic expressions, perfect squares, and expansion formulas. It emphasizes the importance of understanding mathematics beyond just calculations. The author has over 8 years of experience and has mentored thousands of students, including top JEE rank holders.

Uploaded by

Ishan 8th B
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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• 8+ Years Experience

• B.Tech from NITK Surathkal.

• Mentored 8000+ Students &


Top JEE Rank Holders

• 1000+ IITians and counting.

Mathematics is not about numbers,


equations, computations or
algorithms. it is about understanding.

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Factorization

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Type-2

Factorizing the Quadratic Expression by Middle Term splitting.

Example

Factorize: 𝑥2 + 5𝑥 + 6

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Factorize: 𝑥 2 − 14𝑥 + 48

Ans. 𝑥 − 6 (𝑥 − 8)

Solution

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Factorize: 6𝑦 2 − 17𝑦 + 12

Ans. 2𝑦 − 3 (3𝑦 − 4)

Solution

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1
Factorize: 𝑠 2 + 𝑟 2 + 2𝑠𝑟 + 2
+1
4 𝑠+𝑟
2
2 𝑠+𝑟 2+1
Ans.
2(𝑠 + 𝑟)

Solution

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Factorize: 𝑧 2 − 3+ 2 𝑧+ 6

Ans. 𝑧 − 2 𝑧− 3

Solution

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Type-3

Factorization by converting the given expression into a perfect square.

Example

Factorize: 𝑥4 + 4

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Factorize: 𝑎4 + 𝑏4

Ans. 𝑎2 + 𝑏2 − 2 𝑎𝑏
𝑎2 + 𝑏2 + 2 𝑎𝑏

Solution

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Factorize: 𝑥 4 + 𝑥 2 + 1

Ans. (𝑥 2 − 𝑥 + 1)(𝑥 2 + 𝑥 + 1)

Solution

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Factorize: 𝑥 4 + 3𝑥 2 𝑦 2 + 4𝑦 4

Ans. (𝑥 2 + 2𝑦 2 − 𝑥𝑦)(𝑥 2 + 2𝑦 2 + 𝑥𝑦)

Solution

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Factorize: 3𝑥 2 + 4𝑦 2 + 8𝑥𝑦

Ans. (𝑥 + 2𝑦)(3𝑥 + 2𝑦)

Solution

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Factorize: 3𝑥 2 + 6𝑥𝑦 − 2𝑥 + 2𝑦 − 1

Ans. (3𝑥 + 1)(𝑥 + 2𝑦 − 1)

Solution

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Type-4

Factorizing using expansion formula of 𝑎3 − 𝑏3 and 𝑎3 + 𝑏3

𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎 2 − 𝑎𝑏 + 𝑏2 )
𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎 2 + 𝑎𝑏 + 𝑏2 )

Example

Factorize: 𝑥3 + 8

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Factorize: 𝑥 3 + 27𝑦 3

Ans. 𝑥 + 3𝑦 (𝑥 2 + 9𝑦 2 − 3𝑥𝑦)

Solution

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Factorize: 27𝑞 3 − 1

Ans. 3𝑞 − 1 (9𝑞2 + 3𝑞 + 1)

Solution

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Factorize: 𝑎6 − 𝑏6

Ans. 𝑎 − 𝑏 𝑎 + 𝑏 𝑎2 − 𝑎𝑏 + 𝑏2
(𝑎2 + 𝑎𝑏 + 𝑏2)

Solution

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1 2
If, 𝑥 = 1 + 23 + 23 then 𝑥 3 − 3𝑥 2 − 3𝑥 = ?

Ans. 1

Solution

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Find the sum of first 511 terms of

1 1 1
Ans. 7 1 2 + 2 1 2 + 2 1 2 + …….= ?
1 + 23 + 23 23 + 63 + 33 33 + 123 + 43

Solution

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ONLINE
23 CHEMISTRY

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