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Maths 1 Marks Question

Factorisation of Polynomials · Gujarat Board (GSEB) STD 9 (GSEB - English Medium). 49 questions.

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49 questions · timed · auto-graded

Question 51 Mark
Factorise:

3ax - 6ay - 8by + 4bx

Answer
3ax - 6ay - 8by + 4bx
= 3a(x - 2y) + 4b(x - 2y)
= (x - 2y)(3a + 4b)
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Question 61 Mark
Factorise:
a(a + b - c) - bc
Answer
a(a + b - c) - bc
= a2 + ab - ac - bc
= a(a + b) - c(a + b)
= (a - c)(a + b)
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Question 71 Mark
Factorise:
(x + 1)3 + (x - 1)3
Answer
(x + 1)3 + (x - 1)3
= (x + 1 + x - 1)[(x + 1)2 - (x + 1)(x - 1) + (x - 1)2]
= 2x(x2 + 2x + 1 - x2 + 1 + x2 - 2x + 1)
= 2x(x2 + 3)
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Question 81 Mark
Factorise:
a(a - 2b - c) + 2bc
Answer
a(a - 2b - c) + 2bc
= a2 - 2ab - ac + 2bc
= a(a - 2b) - c(a - 2b)
= (a - 2b)(a - c)
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Question 91 Mark
Factorise:
2x + 4y - 8xy - 1
Answer
2x + 4y - 8xy - 1
= 2x - 1 - 8xy + 4y
= (2x - 1) - 4y(2x - 1)
= (2x - 1)(1 - 4y)
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Question 101 Mark
Factorise:
a2 + 2ab + b2 - 9c2
Answer
a2 + 2ab + b2 - 9c2
= (a + b)2 - (3c)2
= (a + b + 3c)(a + b - 3c) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 111 Mark
Factorise:
20x2 - 45
Answer
20x2 - 45
= 5(4x2 - 9)
= 5[(2x)2 - (3)2]
= 2(1 + 5x)(1 - 5x) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= 5(2x + 3)(2x - 3)
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Question 121 Mark
Factorise:
2x(p2 + q2) + 4y(p2 + q2)
Answer
2x(p2 + q2) + 4y(p2 + q2)
= (2x + 4y)(p2 + q2)
= 2(x+ 2y)(p2 + q2)
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Question 131 Mark
Factorise:
8ab2 - 18a3
Answer
8ab2 - 18a3
= 2a(4b2 - 9a2)
= 2a[(2b)2 - (3a)2]
= 2a(2b + 3a)(2b - 3a) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 141 Mark
Factorise:
(3a - 1)2 - 6a + 2
Answer
(3a - 1)2 - 6a + 2
= (3a - 1)2 - 2(3a - 1)
= (3a - 1)[(3a - 1) - 2]
= (3a - 1)(3a - 3)
= 3(3a - 1)(a - 1)
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Question 151 Mark
Factorise:
x3 + 27
Answer
x3 + 27
= x3 + 33
= (x + 3)(x2 - 3x + 9) Since a3 + b = (a + b)(a2 - ab + b2)
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Question 161 Mark
Factorise:
5 - 20x2
Answer
5 - 20x2
= 5(1 - 4x2)
= 5[(1)2 - (2x)2]
= 5[(1 - 2x)(1 + 2x)]
= 5(1 - 2x)(1 + 2x)
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Question 171 Mark
Factorise:
2 - 50x2
Answer
2 - 50x2
= 2(1 - 25x2)
= 2[(1)2 - (5x)2]
= 2(1 + 5x)(1 - 5x) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 181 Mark
Factorise:
64a3 - 27b3 - 144a2b + 108ab2
Answer
64a3 - 27b3 - 144a2b + 108ab2
= (4a)3 - (3b)3 - 3(4a)2(3b) + 3(4a)(3b)2
= (4a - 3b)3
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Question 191 Mark
Factorise:
x3 - 2x2y + 3xy2 - 6y3
Answer
x3 - 2x2y + 3xy2 - 6y3
= x2(x - 2y) + 3y2(x - 2y)
= (x - 2y)(x2 + 3y2)
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Question 201 Mark
Expand:
(a + 2b + 5c)2
Answer
(a + 2b + 5c)2 
= (a)2 + (2b)2 + (5c)2 + 2(a) + 2(2b)(5c) + 2(5c)(a)
= a2 + 4b2 + 25c2 + 4ab + 20bc + 10ac
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Question 211 Mark
Factorise:
27a2 - 48b2
Answer
27a2 - 48b2
= 3(9a2 - 16b2)
= 3[(3a)2 - (4b)2]
= 3(3a + 4b)(3a - 4b)$\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 231 Mark
Factorise:
3x3 - 48x
Answer
3x3 - 48x
= 3x(x2 - 16)
= 3x[(x)2 - (4)2]
= 3x(x + 4)(x - 4)$\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 241 Mark
Factorise:
2a2 + bc - 2ab - ac
Answer
2a2 + bc - 2ab - ac
= 2a2 - 2ab - ac + bc
= 2a(a - b) - c(a - b)
= (a - b)(2a - c)
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Question 251 Mark
Factorise:

a3b - a2b + 5ab - 5b

Answer
a3b - a2b + 5ab - 5b
= a2b(a - 1) + 5b(a - 1)
= (a - 1)(a2b + 5b)
= (a - 1)b(a2 + 5)
= b(a - 1)(a2 + 5)
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Question 281 Mark
Factorise:
a3x3 - 3a2bx2 + 3ab2 x - b3
Answer
a3x3 - 3a2bx2 + 3ab2 x - b3
(ax)3 - (b)3 - 3(ax)2(b) +(ax)(b)2
= (ax - b)3
Hence, factorisation of a3x3 - 3a2bx2 + 3ab2 x - b3 is = (ax - b)3
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Question 291 Mark
Factorise:
a3 - 12a(a - 4) - 64
Answer
a3 - 12a(a - 4) - 64
a3 - 12a2 + 48a - 64
= (a)3 - (4)3 - 3(a)2(4) + 3(a)(4)2
= (a - 4)3
Hence, factorisation of a3 - 12a(a - 4) - 64 is = (a - 4)3
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Question 301 Mark
Factorise:
$\Big(\frac{25}{4}\text{x}^2-\frac{1}{9}\text{y}^2\Big)$
Answer
$\Big(\frac{25}{4}\text{x}^2-\frac{1}{9}\text{y}^2\Big)$
$\Big(\frac{5}{2}\text{x}\Big)^2-\Big(\frac{1}{3}\text{y}\Big)^2$
$=\Big(\frac{5}{2}\text{x}-\frac{1}{3}\text{y}\Big)\Big(\frac{5}{2}\text{x}+\frac{1}{3}\text{y}\Big)$
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Question 321 Mark
Factorise:
150 - 6x2
Answer
150 - 6x2
= 6(25 - x2)
= 6(52 - x2)
= 6(5 + x)(5 - x) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 331 Mark
Factorise:
a3 + ab(1 - 2a) - 2b2
Answer
a3 + ab(1 - 2a) - 2b2
= a3 + ab - 2a2b - 2b2
= a(a2 + b) - 2b(a2 + b)
= (a2 + b)(a - 2b)
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Question 341 Mark
Factorise:
125x3 - 27y3 - 225x2y + 135xy2
Answer
125x3 - 27y3 - 225x2y + 135xy2
(5x)3 - (3y)3 - 3(5x)2(3y)+ 3(5x)(3y)2
= (5x - 3y)3
Hence, factorisation of 125x3 - 27y3 - 225x2y + 135xy2 is (5x - 3y)3
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Question 371 Mark
Factorise:
a6 + b6
Answer
a6 + b6
= (a2)3 + (b2)3
= (a2 + b2)[(a2)2 - (a2b2) + (b2)2]
= (a2 + b2)(a4 - a2b2 + b4)
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Question 381 Mark
Factorise:
a2 - b2 - a - b
Answer
a2 - b2 - a - b
= a2 - b2 - (a + b)
= (a - b)(a + b) - (a + b)
= (a + b)(a - b - 1)
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Question 391 Mark
Factorise:
8a3 + 27b3 + 36a2b + 54ab2
Answer
8a3 + 27b3 + 36a2b + 54ab2
= (2a)3 + (3b)3 + 3(2a)2(3b) + 3(2a)(3b)2
= (2a + 3b)3
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Question 401 Mark
Factorise:
$\frac{64}{125}\text{a}^3-\frac{96}{25}\text{a}^2+\frac{48}{5}\text{a}-8$
Answer
$\frac{64}{125}\text{a}^3-\frac{96}{25}\text{a}^2+\frac{48}{5}\text{a}-8$
$\Big(\frac{4}{5}\text{a}\Big)^3-(2)^3-3\Big(\frac{4}{5}\text{a}\Big)^2(2)+3\Big(\frac{4}{5}\text{a}\Big)(2)^2$
$=\Big(\frac{4}{5}\text{a}-2\Big)^3$
Hence, factorisatoin of $\frac{64}{125}\text{a}^3-\frac{96}{25}\text{a}^2+\frac{48}{5}\text{a}-8$ is $\Big(\frac{4}{5}\text{a}-2\Big)^3$
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Question 411 Mark
Factorise:
(2x - 3)2 - 8x + 12
Answer
(2x - 3)2 - 8x + 12
= (2x - 3)2 - 4(2x - 3)
= (2x - 3)(2x - 3 - 4)
= (2x - 3)(2x - 7)
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Question 421 Mark
Factorise:
x - 64x3
Answer
x - 64x3
= x(1 - 64x2)
= x[(1)2 - (8x)2]
= x(1 + 8x)(1 - 8x) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 441 Mark
Factorise:
abx2 + a2x + b2x + ab
Answer
abx2 + a2x + b2x + ab
= ax(bx + a) + b(bx + a)
= (bx + a)(ax + b)
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Question 451 Mark
Factorise:
x3 - 5x2 - x + 5
Answer
x3 - 5x2 - x + 5
= x2(x - 5) - 1(x - 5)
= (x - 5)(x2 - 1) $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
= (x - 5)(x + 1)(x - 1)
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Question 461 Mark
Factorise:
27a3 + 64b3
Answer
27a3 + 64b3
= (3a)3 + (4b)3
= (3a + 4b)[(3a)2 - (3a)(4b) + (4b)2]
= (3a + 4b)(9a2 - 12ab + 16b2)
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Question 471 Mark
Factorise:
(3a + 5b)2 - 4c2
Answer
(3a + 5b)2 - 4c2
= (3a + 5b)2 - (2c)2
= (3a + 5b - 2c)(3a + 5b + 2c)
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