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MATHS 1 Marks Question

Factorisation of Polynomials · Rajasthan Board (RBSE) STD 9 (Rajasthan - English Medium). 49 questions.

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49 questions · timed · auto-graded

Question 11 Mark
Factorise:$a^3b - a^2b + 5ab - 5b$
Answer
$a^3b - a^2b + 5ab - 5b$
$= a^2b(a - 1) + 5b(a - 1)$
$= (a - 1)(a^2b + 5b)$
$= (a - 1)b(a^2 + 5)$
$= b(a - 1)(a^2 + 5)$
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Question 21 Mark
Factorise: $2a^2 + bc - 2ab - ac$
Answer
$2a^2 + bc - 2ab - ac$
$= 2a^2 - 2ab - ac + bc$
$= 2a(a - b) - c(a - b)$
$= (a - b)(2a - c)$
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Question 61 Mark
Factorise: $8 - 4a - 2a^3 + a^4$
Answer
$8 - 4a - 2a^3 + a^4$
$= 4(2 - a) - a^3(2 - a)$
$= (2 - a)(4 - a^3)$
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Question 71 Mark
Factorise: $a(a + b - c) - bc$
Answer
$a(a + b - c) - bc$
$= a^2 + ab - ac - bc$
$= a(a + b) - c(a + b)$
$= (a - c)(a + b)$
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Question 81 Mark
Factorise:$(x + 1)^3 + (x - 1)^3$
Answer
$(x + 1)^3 + (x - 1)^3$
$= (x + 1 + x - 1)[(x + 1)^2 - (x + 1)(x - 1) + (x - 1)^2]$
$= 2x(x^2 + 2x + 1 - x^{2 }+ 1 + x^2 - 2x + 1)$
$= 2x(x^2 + 3)$
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Question 91 Mark
Factorise: $a(a - 2b - c) + 2bc$
Answer
$a(a - 2b - c) + 2bc$
$= a^2 - 2ab - ac + 2bc$
$= a(a - 2b) - c(a - 2b)$
$= (a - 2b)(a - c)$
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Question 101 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 111 Mark
Factorise: $a^2 + 2ab + b^2 - 9c^2$
Answer
$a^2 + 2ab + b^2 - 9c^2$
$= (a + b)^2 - (3c)^2$
$= (a + b + 3c)(a + b - 3c)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 121 Mark
Factorise: $20x^2 - 45$
Answer
$20x^2 - 45$
$= 5(4x^2 - 9)$
$= 5[(2x)^2 - (3)^2]$
$= 2(1 + 5x)(1 - 5x)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
$= 5(2x + 3)(2x - 3)$
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Question 131 Mark
Factorise: $2x(p^2 + q^2) + 4y(p^2 + q^2)$
Answer
$2x(p^2 + q^2) + 4y(p^2 + q^2)$
$= (2x + 4y)(p^2 + q^2)$
$= 2(x+ 2y)(p^2 + q^2)$
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Question 141 Mark
Factorise:$8ab^2 - 18a^3$
Answer
$8ab^2 - 18a^3$
$= 2a(4b^2 - 9a^2)$
$= 2a[(2b)^2 - (3a)^2]$
$= 2a(2b + 3a)(2b - 3a)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 151 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 161 Mark
Factorise: $x^3 + 27$
Answer
$x^3 + 27$
$= x^3 + 3^3$
$= (x + 3)(x^2 - 3x + 9)$ Since $a^3 + b = (a + b)(a^2 - ab + b^2)$
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Question 171 Mark
Factorise:$5 - 20x^2$
Answer
$5 - 20x^2$
$= 5(1 - 4x^2)$
$= 5[(1)^2 - (2x)^2]$
$= 5[(1 - 2x)(1 + 2x)]$
$= 5(1 - 2x)(1 + 2x)$
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Question 181 Mark
Factorise:$2 - 50x^2$
Answer
$2 - 50x^2$
$= 2(1 - 25x^2)$
$= 2[(1)^2 - (5x)^2]$
$= 2(1 + 5x)(1 - 5x)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 191 Mark
Factorise:$64a^3 - 27b^3 - 144a^2b + 108ab^2$
Answer
$64a^3 - 27b^3 - 144a^2b + 108ab^2$
$= (4a)^3 - (3b)^3 - 3(4a)^2(3b) + 3(4a)(3b)^2$
$= (4a - 3b)^3$
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Question 201 Mark
Factorise:$x^3 - 2x^2y + 3xy^2 - 6y^3$
Answer
$x^3 - 2x^2y + 3xy^2 - 6y^3$
$= x^2(x - 2y) + 3y^2(x - 2y)$
$= (x - 2y)(x^2 + 3y^2)$
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Question 211 Mark
Expand:$(a + 2b + 5c)^2$
Answer
$(a + 2b + 5c)^2$
$= (a)^2 + (2b)^2 + (5c)^2 + 2(a) + 2(2b)(5c) + 2(5c)(a)$
$= a^2 + 4b^2 + 25c^2 + 4ab + 20bc + 10ac$
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Question 221 Mark
Factorise:$27a^{2 }- 48b^2$
Answer
$27a^{2 }- 48b^2$
$= 3(9a^2 - 16b^2)$
$= 3[(3a)^2 - (4b)^2]$
$= 3(3a + 4b)(3a - 4b)\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 241 Mark
Factorise:$3x^3 - 48x$
Answer
$3x^3 - 48x$
$= 3x(x^2 - 16)$
$= 3x[(x)^2 - (4)^2]$
$= 3x(x + 4)(x - 4)\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 261 Mark
Factorise:$a^3x^3 - 3a^2bx^2 + 3ab^2 x - b^3$
Answer
$a^3x^3 - 3a^2bx^2 + 3ab^2 x - b^3$
$(ax)^3 - (b)^3 - 3(ax)^2(b) +(ax)(b)^2$
$= (ax - b)^3$
Hence, factorisation of $a^3x^3 - 3a^2bx^2 + 3ab^2 x - b^3$​​​​​​​ is $= (ax - b)^3$
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Question 271 Mark
Factorise:$a^3 - 12a(a - 4) - 64$
Answer
$a^3 - 12a(a - 4) - 64$
$a^3 - 12a^2 + 48a - 64$
$= (a)^3 - (4)^3 - 3(a)^2(4) + 3(a)(4)^2$
$= (a - 4)^3$
Hence, factorisation of $a^3 - 12a(a - 4) - 64$ is $= (a - 4)^3$​​​​​​​
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Question 281 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 291 Mark
Factorise:$x^3 + 2x^2 + 5x + 10$
Answer
$x^3 + 2x^2 + 5x + 10$
$= x^2(x + 2) + 5(x + 2)$
$= (x^2 + 5)(x + 2)$
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Question 301 Mark
Factorise:$150 - 6x^2$
Answer
$150 - 6x^2$
$= 6(25 - x^2)$
$= 6(5^2 - x^2)$
$= 6(5 + x)(5 - x)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 311 Mark
Factorise:$a^3 + ab(1 - 2a) - 2b^2$
Answer
$a^3 + ab(1 - 2a) - 2b^2$
$= a^3 + ab - 2a^2b - 2b^2$
$= a(a^2 + b) - 2b(a^2 + b)$
$= (a^2 + b)(a - 2b)$
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Question 321 Mark
Factorise:$125x^3 - 27y^3 - 225x^2y + 135xy^2$
Answer
$125x^3 - 27y^3 - 225x^2y + 135xy^2$
$(5x)^3 - (3y)^3 - 3(5x)^2(3y)+ 3(5x)(3y)^2$
$= (5x - 3y)^3$
Hence, factorisation of $125x^3 - 27y^3 - 225x^2y + 135xy^2 is (5x - 3y)^3$​​​​​​​
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Question 351 Mark
Factorise:$a^6 + b^6$
Answer
$a^6 + b^6$
$= (a^2)^3 + (b^2)^3$
$= (a^2 + b^2)[(a^2)^2 - (a^2b^2) + (b^2)^2]$
$= (a^2 + b^2)(a^{4 }- a^2b^2 + b^4)$
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Question 361 Mark
Factorise:$a^2 - b^2 - a - b$
Answer
$a^2 - b^2 - a - b$
$= a^2 - b^2 - (a + b)$
$= (a - b)(a + b) - (a + b)$
$= (a + b)(a - b - 1)$
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Question 371 Mark
Factorise:$8a^3 + 27b^3 + 36a^2b + 54ab^2$
Answer
$8a^3 + 27b^3 + 36a^2b + 54ab^2$
$= (2a)^3 + (3b)^3 + 3(2a)^2(3b) + 3(2a)(3b)^2$
$= (2a + 3b)^3$
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Question 381 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 391 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 401 Mark
Factorise:$x - 64x^3$
Answer
$x - 64x^3$
$= x(1 - 64x^2)$
$= x[(1)^2 - (8x)^2]$
$= x(1 + 8x)(1 - 8x)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
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Question 421 Mark
Factorise: $abx^2 + a^2x + b^2x + ab$
Answer
$abx^2 + a^2x + b^2x + ab$
$= ax(bx + a) + b(bx + a)$
$= (bx + a)(ax + b)$
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Question 431 Mark
Factorise:$x^3 - 5x^2 - x + 5$
Answer
$x^3 - 5x^2 - x + 5$
$= x^2(x - 5) - 1(x - 5)$
$= (x - 5)(x^2 - 1)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
$= (x - 5)(x + 1)(x - 1)$
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Question 441 Mark
Factorise: $27a^3 + 64b^3$
Answer
$27a^3 + 64b^3$
$= (3a)^3 + (4b)^3$
$= (3a + 4b)[(3a)^2 - (3a)(4b) + (4b)^2]$
$= (3a + 4b)(9a^2 - 12ab + 16b^2)$
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Question 451 Mark
Factorise:$(3a + 5b)^2 - 4c^2$
Answer
$(3a + 5b)^2 - 4c^2$
$= (3a + 5b)^2 - (2c)^2$
$= (3a + 5b - 2c)(3a + 5b + 2c)$
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Question 461 Mark
Factorise:$x^2 + xy - 2xz - 2yz$
Answer
$x^2 + xy - 2xz - 2yz$
$= x(x + y) - 2z(x + y)$
$= (x+ y)(x - 2z)$
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Question 481 Mark
Factorise: $a^3 + a - 3a^2 - 3$
Answer
$a^3 + a - 3a^2 - 3$
$= a(a^2 + 1) - 3(a^2 + 1)$
$= (a - 3)(a^2 + 1)$
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Question 491 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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