Mathematics Questions and Answers – Factorisation of Polynomials and Algebraic Identities

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Factorisation of Polynomials and Algebraic Identities”.

1. x-1 is a factor of 4x2-9x-6.
a) True
b) False
View Answer

Answer: b
Explanation: According to factor theorem, x-a is a factor of p(x) if p(a) = 0.
Therefore x-1 is a factor of 4x2-9x-6 is a factor if p(1)=0.
p(1) = 4(1)2-9(1)-6 = 4 – 9 – 6
= -11 ≠ 0
Therefore, we can say that x-1 is not a factor of 4x2-9x-6.
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2. Find the value of k, if x-3 is a factor of 5x3-2x2+x+k.
a) 50
b) 60
c) -60
d) -120
View Answer

Answer: d
Explanation: According to factor theorem, x-a is a factor of p(x) if p(a) = 0.
Here, it is given that x-3 is a factor of 5x3-2x2+x+k.
Therefore, p(3) must be equal to zero.
p(3) = 5(3)3-2(3)2+3+k = 0
Therefore, 5(27) – 2(9) + 3 + k = 0
135 – 18 + 3 + k=0
120 + k = 0
Therefore, k= -120

3. What do we get after factorising x2+6x-27?
a) (x+9)(x-3)
b) (x-9)(x+3)
c) (x-9)(x-3)
d) (x+9)(x+3)
View Answer

Answer: a
Explanation: To factorise x2+6x-27, we have to find two numbers ‘a’ and ‘b’ such that a+b=6 and a*b=27.
For that we have to find factors of -27, which are ±1, ±3, ±9.
Now we have to arrange two numbers from these numbers such that a+b=6 and a*b=27.
By considering this, we get two numbers +9 and -3
9 + (-3) = 6 and 9*-3 = -27
Now after manipulating terms, we get x2+9x- 3x-27.
x2+9x- 3x-27 = x(x+9)-3(x+9)
= (x+9)(x-3).
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4. What do we get after factoring 49x2-28xy+.4y2?
a) (7x+2y)2
b) (49x-4y)2
c) (7x-2y)2
d) (7x-28y)2
View Answer

Answer: c
Explanation: We know that a2-2ab+.b2 = (a-b)2
49x2-28xy+.4y2 can also be written as (7x)2-2(7)(2)xy+(2y)2
Here, a = 7x and b = 2y.
Therefore, 49x2-28xy+.4y2 = (7x)2-2(7)(2)xy+(2y)2
= (7x-2y)2.

5. 26*34 = __________ (calculate without direct calculation).
a) 900
b) 884
c) 916
d) 844
View Answer

Answer: b
Explanation: 26*34 can also be written as (30-4)*(30+4)
We know that (a-b)*(a+b) = a2 -b2
Similarly, 26*34 = (30-4)*(30+4)
= 302 – 42
= 900 – 16
= 884.
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6. 27*29 = __________ (calculate without direct calculation).
a) 783
b) 753
c) 763
d) 793
View Answer

Answer: a
Explanation: We know that (a+b)*(a+c) = a2 + (b+c)a + bc
27*29 can also be written as (25+2)*(25+4)
Now using above identity, 27*29 = (25+2)*(25+4)
= 252 + (2+4)25 + (4)(2)
= 625 + 6(25) + 8
= 625 + 150 + 8
= 783.

7. 953 = __________ (calculate without direct calculation).
a) 856395
b) 857625
c) 857375
d) 852395
View Answer

Answer: c
Explanation: We know that (x-y)3 = x3– y3 – 3 xy (x-y).
953 can also be written as (100-5)3
Therefore, 953= (100-5)3 = (100)3 – (5)3 – 3(100)(5)(100-5)
= 1000000 – 125 – 1500(95)
= 1000000 – 125 – 142500
= 857375.

8. What do we get after expanding (p+3q-2z)2?
a) p2 + 9q2 + 4z2 + 6pq – 12qz + 4zp
b) p2 + 9q2 + 4z2 + 12pq + 12qz + 4zp
c) p2 + 9q2 + 4z2 – 12pq – 12qz – 4zp
d) p2 + 9q2 + 4z2 + 6pq – 12qz – 4zp
View Answer

Answer: d
Explanation: We know that (a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(p+3q-2z)2can also be written as (p+3q+(-2z))2.
Here, a = p, b = 3q and c = -2z
Therefore, using that formula we get (p+3q+(-2z))2
= p2 + (3q)2 + (-2z)2 + 2(p)(3q) + 2(3q)(-2z) + 2(-2z)(p)
= p2 + 9q2 + 4z2 + 6pq – 12qz – 4zp.
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9. What do we get after factorising x3+ 8y3+ z3 – 4xyz?
a) (x + 2y + z) (x2 + 4y2 + z2 – 2xy – 2yz – zx)
b) (x + 2y + z) (x2 + 4y2 + z2 – 2xy – 2yz – zx)
c) (x + 2y + z) (x2 + 4y2 + z2 – 2xy – 2yz – zx)
d) (x + 2y + z) (x2 + 4y2+ z2 – 2xy – 2yz – zx)
View Answer

Answer: c
Explanation: We know that a3 + b3 + c3 – 3abc = (a+b-c)(a2 + b2 + c2 – ab – bc – ca)
In x3 + 8y3 + y3 – 6xyz, a=x, b=2y and c=z
By using the above equation, we get x3 + 8y3 + z3 – 4xyz = (x+2y+z) (x2 + (2y)2 + z2 – x(2y) – (2y)(z) – zx)
= (x + 2y + z) (x2 + 4y2 + z2 – 2xy – 2yz – zx).
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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter