FACTORISATION
Q1] Factorise the following: [14]
a] (i) 20m¹⁰- 15m⁵p ⁶
(ii) lm² - mn² - lm + n²
(iii) 4x²- 9 y²
(iv) a² - 6ab + 9b²
(v) x⁴ - 256
(vi) (3x _ 2y )² _ 5 (3x _ 2y ) _ 24
(vii) 3x²_ 2x _8
Answers
Required Answers:-
(i) 20m¹⁰ - 15m⁵p⁶
→ We can see here in both the terms 5m⁵ is common.
Hence,
Let us take our 5m⁵ from both the terms,
= 5m⁵(4m⁵ - 3p⁶)
Hence, Factorised form of 20m¹⁰ - 15m⁵p⁶ is 5m⁵(4m⁵ - 3p⁶)
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(ii) Im² - mn² - Im + n²
→ Let us bracket pair of terms.
(Im² - mn²) - (Im - n²)
Here we can see there are two terms in both the terms we will take common variables out.
= m(Im - n²) - 1(Im - n²)
Now,
(Im - n²) is common, so we'll take out it as common.
= (Im - n²)(m - 1)
Hence, Factorised form of Im² - mn² - Im + n² is (Im - n²)(m - 1)
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(iii) 4x² - 9y²
→ Here 4x² is the square of 2x and 9y² is the square of 3y.
Hence,
4x² - 9y² = (2x)² - (3y)²
Applying the identity:-
- a² - b² = (a + b)(a - b)
= (2x)² - (3y)²
= (2x + 3y)(2x - 3y)
Hence, the Factorised form of 4x² - 9y² is (2x + 3y)(2x - 3y)
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(iv) a² - 6ab + 9b²
→ Let us expand the given identity.
= (a)² - 2 × a × 3b + (3b)²
We know,
- (a - b)² = a² - 2 × a × b + b²
Hence,
(a)² - 2 × a × 3b + (3b)²
= (a - 3b)²
= (a - 3b)(a - 3b)
Hence, the Factorised form of a² - 6ab + 9b² is (a - 3b)(a - 3b)
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(v) x⁴ - 256
→ Here x⁴ is the square of x² and 256 is the square of 16
Hence,
x⁴ - 256 = (x²)² - (16)²
Applying the identity:-
- a² - b² = (a + b)(a - b)
= (x² + 16)(x² - 16)
Now,
Factorising the second bracket again by the same identity.
= (x² + 16)[(x)² - (4)²]
= (x² + 16)(x + 4)(x - 4)
Hence, the Factorised form of x⁴ - 256 is (x² + 16)(x + 4)(x - 4).
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(vi) (3x - 2y)² - 5(3x - 2y) - 24
→ Let (3x - 2y) = a
Hence,
a² - 5a - 24
By splitting the middle term,
= a² - 8a + 3a - 24
= a(a - 8) + 3(a - 8)
Taking (a - 8) common,
= (a - 8)(a + 3)
Now
Putting the value of a
= (3x - 2y - 8)(3x - 2y + 3)
Hence the Factorised form of (3x - 2y)² - 5(3x - 2y) - 24 is (3x - 2y - 8)(3x - 2y + 3).
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(vii) 3x² - 2x - 8
By splitting the middle term,
= 3x² - 6x + 4x - 8
= 3x(x - 2) + 4(x - 2)
Taking (x - 2) as common.
= (x - 2)(3x + 4)
Hence, the Factorised form of (x - 2)(3x + 4).
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Some other identities:-
- (a + b)² = a² + 2ab + b²
- (a + b)³ = a³ + b³ + 3a²b + 3ab²
- (a - b)³ = a³ - b³ + 3a²b - 3ab²
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