Edudel, Directorate of Education Govt. of NCT of Delhi Class 8 Mental Maths Question Bank Chapter 14 Factorization Questions Solution. In this chapter, there are total 50 math problems. We have given each questions solution step by step.
Edudel Class 8 Mental Maths Chapter 14 Factorization:
1.) What is the degree of algebraic expression y3 + 2y + 8
ANSWER:
We Know,
The degree of a polynomial is the highest power of its variables.
The degree of algebraic expression y3 + 2y + 8 is 3.
2.) If a + 1/a = √3 what is the value of a 2 + 1/ (a 2)
ANSWER:
We know,
(a + b) 2 = a2 + b2 + 2ab
a = a b = 1/a
(a + 1/a) 2 = a2 + 1/ (a 2) + 2a x 1/a
(a + 1/a) 2 = a2 + 1/ (a 2) + 2
a2 + 1/ (a 2) = (a + 1/a) 2 – 2
a2 + 1/ (a 2) = 3 – 2
a2 + 1/ (a 2) = 1
3.) What is the quotient, when we divide 17xy by xy?
ANSWER:
When we divide,
17xy / xy
The quotient we get is 17.
4.) If 4ab is one of the factor of (4a 2 x b + 4a x b 2), what is the other factor?
ANSWER:
Given that,
4ab is one of the factor of (4a 2 x b + 4a x b 2)
We have to find other factor.
We know,
4ab x (a + b) = (4a 2 x b + 4a x b 2)
The other factor is (a + b).
5.) What is the HCF of 25x 2y 2, 65x 4y 3, 95x 10y 4?
ANSWER:
We have to find HCF of 25x 2y 2, 65x 4y 3, 95x 10y 4
HCF is the highest common factor of the given expressions.
25x 2y 2, 65x 4y 3, 95x 10y 4 from this highest common factor is 5x 2y 2
HCF of 25x 2y 2, 65x 4y 3, 95x 10y 4 is 5x 2y 2.
6.) What is the degree of the quotient of (48x 4 – 12y4)/60?
ANSWER:
We have to find the degree of the quotient of (48x 4 – 12y4)/60
12 (4 x 4 – y4) / 60
= (4 x 4 – y4) / 5
We know,
The degree of a polynomial is the highest power of its variables.
The degree of the quotient is 4.
7.) What is the common factor of uv + 9u and 2(v + 3)?
ANSWER:
We have to find the common factor of uv + 9u and 2(v + 3)
The common factor of uv + 9u and 2(v + 3) is (v + 3).
- Factorise: k 6 – 12k3
ANSWER:
We have to Factorise: k 6 – 12k3
We take k3 as common factor.
k3(k3 – 12)
Factorisation of k 6 – 12k3 is k3 (k3 – 12)
9.) Simplify: (77xyz)/ (7x)
ANSWER:
We have to simplify (77xyz)/ (7x)
We divide (77xyz) by (7x)
(77xyz)/ (7x) = 11yz
10.) What should be added to 16x2 + 9 to make it (4x – 3)2?
ANSWER:
We have to find what should be added to 16x2 + 9 to make it (4x – 3)2
We know,
(a – b)2 = a2 + b2 – 2ab
(4x – 3)2 = (4x) 2 + 32 – 24x
(4x – 3)2 =16x2 + 9 – 24x
We add – 24x in 16x2 + 9 to make it (4x – 3)2
11.) Which identity will be used to factorise a2 + 20b + 100?
ANSWER:
a2 + 20b + 100
Here we use,
(a + b)2 = a2 + b2 + 2ab
(a + 10)2 = a2 + 102+ 2 x a x 10
(a + 10)2 = a2 + 100 + 20b
Identity we use is (a + b) 2 = a2 + b2 + 2ab.
12.) Find the value of (105 x 105 – 5 x 5)
ANSWER:
(105 x 105 – 5 x 5)
This is in the form of (a2 – b2)
We know,
(a2 – b2) = (a + b) (a – b)
(1052 – 52) = (105 + 5) (105 – 5)
(1052 – 52) = 110 x 100
(1052 – 52) = 11000
13.) What will be the product of (3x + 2) (3x – 2)?
ANSWER:
We have to find the product of (3x + 2) (3x – 2)
This is in the form of (a + b) (a – b)
We know,
(a2 – b2) = (a + b) (a – b)
(3x + 2) (3x – 2) = (3x2 – 22)
(3x2 – 22) = 9x2 – 4
14.) What will be the quotient on dividing x 2 – x – 30 by (x – 6)?
ANSWER:
We have to find the quotient on dividing x 2 – x – 30 by (x – 6)
x2 – x – 30 / (x – 6)
Dividing we get,
The quotient is x + 5
15.) What will be the common factor of (u + v) (a + b) and w (a + b)?
ANSWER:
We have to find the common factor of (u + v) (a + b) and w (a + b)
The common factor of (u + v) (a + b) and w (a + b) is (a + b)
16.) What should be added to the left hand side to rectify the equation 4x + 2 = 4(x + 2)
ANSWER:
4x + 2 = 4(x + 2)
We have to find which number added to the left hand side to rectify the equation
4x + 2 = 4x + 8
We have to add 6 to the left hand side to rectify the equation.
17.) What is the remainder when divisor is a factor of the dividend?
ANSWER:
When divisor is a factor of the dividend, the remainder is 0.
- Factorising 100x 4 – 81y 4 gives 10x 2 – 9y 2 as one factor, what is the other factor?
ANSWER:
100x 4 – 81y 4
One factor is 10x 2 – 9y 2 we have to find other factor
10x 2 – 9y 2 x other factor = 100x 4 – 81y 4
Other factor = 100x 4 – 81y 4 / 10x 2 – 9y 2
Other factor =10x 2 + 9y 2
19.) Simplify: (x 4 – 16)/ ((x 2 + 4) (x – 2))
ANSWER:
We have to simplify (x 4 – 16)/ ((x 2 + 4) (x – 2))
(x4 – 16) = (x 2)2 – 42
This is in the form of (a2 – b2)
We know,
(a2 – b2) = (a + b) (a – b)
(x 2)2 – 42 = ((x 2) + 4) ((x 2) – 4)
(x4 – 16) = ((x 2) + 4) ((x 2) – 4)
We put (x 4 – 16) as ((x 2) + 4) ((x 2) – 4)
(x4 – 16)/ ((x 2 + 4) (x – 2)) = ((x 2) + 4) ((x 2) – 4) / ((x 2 + 4) (x – 2))
We get,
((x 2) + 4) ((x 2) – 4) / ((x 2 + 4) (x – 2)) = x + 2
20.) Find the value of (105) 2
ANSWER:
(105) 2
105 x 105
(105) 2 = 11025
21.) Evaluate: 99 x 101
ANSWER:
We have to find 99 x 101
We write this as
(100 – 1) x (100 + 1)
This is in the form of (a + b) (a – b)
We know,
(a2 – b2) = (a + b) (a – b)
(100 – 1) x (100 + 1) = (1002 – 12)
10000 – 1
99 x 101 = 9999
- ) Factorise: x(y – z) + y(y – z)
ANSWER:
x(y – z) + y(y – z)
In this (y – z) is common factor. We take outside.
x(y – z) + y(y – z) = (y – z) (x + y)
23.) If a = 6 b = 5 find a 2 – b 2
ANSWER:
We know,
(a2 – b2) = (a + b) (a – b)
a = 6 b = 5
(62 – 52) = (6 + 5) (6 – 5)
(62 – 52) = 11 x 1
(62 – 52) = 11
24.) What will be the coefficient of u 2 in the quotient of algebraic expression (3u 3 + 5u 2 + 7)/ (u + 2)
ANSWER:
Given algebraic expression (3u 3 + 5u 2 + 7)/ (u + 2)
We have to find the coefficient of u 2 in the quotient of algebraic expression (3u 3 + 5u 2 + 7)/ (u + 2)
The coefficient of u 2 in the quotient of algebraic expression (3u 3 + 5u 2 + 7)/ (u + 2) is 3.
25.) What will be the quotient of algebraic expression (4x2y + 8x2y2 – 16xy2)/ (4xy)?
ANSWER:
Given,
Algebraic expression (4x2y + 8x2y2 – 16xy2)/ (4xy)
We have to find the quotient of algebraic expression
(4x2y + 8x2y2 – 16xy2)/ (4xy) = x + 2xy -4y
The quotient of algebraic expression is x + 2xy -4y
26.) What will be the coefficient of a 4 in the product of (1/4 a 2 + b 2) (a 2 – 3/2 b2)
ANSWER:
(1/4 a 2 + b 2) (a2 – 3/2 b2)
= 1/4 a 2(a 2 – 3/2 b2) + b 2 (a 2 – 3/2 b2)
= 1/4 a 4 – 3/8 a 2 b2 + a 2 b2 – 3/2 b4
The coefficient of a 4 is 1/4.
27.) The area of a playground is (14p2 – 35p) square units and one of its side’s measures 7p units. What is the measure of the other side?
ANSWER:
Given that, the area of a playground is (14p2 – 35p) square units
One of its side’s measures 7p units
We have to find other side.
7p x other side of a playground = (14p2 – 35p) square units
Other side of a playground = (14p2 – 35p) square units / 7p
Other side of a playground = (2p – 5) units
28.) If (49x2 + 14x + 35) kilograms of sugar is stored in 7 bags in equal quantities, how many kilograms of sugar is there in each bag?
ANSWER:
Given, (49x2 + 14x + 35) kilograms of sugar is stored in 7 bags
We have to find how many kilograms of sugar is there in each bag.
Sugar in each bag = (49x2 + 14x + 35) / 7
Sugar in each bag = (7x2 + 2x + 5) kilograms
29.) What will be the constant term in the product of (z + 3) (z – 7)?
ANSWER:
We first find product of (z + 3) (z – 7)
z(z – 7) + 3(z – 7)
z2 – 7z + 3z – 21
(z + 3) (z – 7) = z2 – 4z– 21
The constant term in the product of (z + 3) (z – 7) is – 21.
30.) Find a 2 – b 2 if a = – 3 and b = 3
ANSWER:
We know,
(a2 – b2) = (a + b) (a – b)
a = -3 b = 3
(-32 – 32) = (-3 + 3) (-3 – 3)
(-32 – 32) = 0 x -6
(-32 – 32) = 0
31.) Find (11 x 11 – 9 x 9)
ANSWER:
(11 x 11 – 9 x 9)
We write this as (112 – 92)
We know,
(a2 – b2) = (a + b) (a – b)
(112 – 92) = (11 + 9) (11 – 9)
(112 – 92) = 20 x 2
(112 – 92) = 40
32.) Find k if k (a 2 – b 2) = a 4 – b 4
ANSWER:
k (a 2 – b 2) = a 4 – b 4
k = a 4 – b 4 / (a 2 – b 2)
a4 – b 4 = ((a 2) 2 – (b 2) 2)
We know,
(a2 – b2) = (a + b) (a – b)
a4 – b 4 = (a 2 – b 2) (a 2 + b 2)
k = a 4 – b 4 / (a 2 – b 2) = (a 2 – b 2) (a 2 + b 2) / (a 2 – b 2)
k = (a 2 + b 2)
33.) Find q if q (a 2 + b 2) = a 4 – b 4
ANSWER:
q (a2 + b 2) = a 4 – b 4
q = a 4 – b 4 / (a 2 + b 2)
We know,
(a2 – b2) = (a + b) (a – b)
a4 – b 4 = (a 2 – b 2) (a 2 + b 2)
We put a 4 – b 4 = (a 2 – b 2) (a 2 + b 2)
q = a 4 – b 4 / (a 2 + b 2) = (a 2 – b 2) (a 2 + b 2) / (a 2 + b 2)
q = (a 2 – b 2)
34.) What are the prime factors of 45?
ANSWER:
The prime factors of 45 are 3 and 5.
35.) Simplify: 6 2 – 2 x 6 x 5 + 5 2
ANSWER:
6 2 – 2 x 6 x 5 + 5 2
This is in the form of
(a – b)2 = a2 + b2 – 2ab
Here, a = 6 and b = 5
(6 – 5)2 = 62 + 52 – 2 x 6 x 5
(6 – 5)2 = 36 + 25 – 60
(6 – 5)2 = 61 – 60
(6 – 5)2 = 1.
36.) What will be the two numbers P and Q such that P – Q = 2 and PQ = 15?
ANSWER:
Given,
P – Q = 2 and PQ = 15
We know,
(a – b)2 = a2 + b2 – 2ab
(P – Q) 2 = P2 + Q2 – 2PQ
22 = P2 + Q2 – 2 x 15
4 = P2 + Q2 – 30
P2 + Q2= 4 + 30
P2 + Q2 = 34
P – Q = 2
P = 2 + Q
We put P = 2 + Q in P2 + Q2 = 34
(2 + Q) 2 + Q2 = 34
Solving,
We get P = 5 and Q = 3
37.) Evaluate: (2.5) 2 – (1.5) 2
ANSWER:
(2.5) 2 – (1.5) 2
This in form of (a2 – b2)
We know,
(a2 – b2) = (a + b) (a – b)
(2.5) 2 – (1.5) 2 = (2.5 + 1.5) x (2.5 – 1.5)
(2.5) 2 – (1.5) 2 = 4 x 1
(2.5) 2 – (1.5) 2 = 4
38.) If 2a + 3b = 12 and 2a – 3b = 20, find a.
ANSWER:
2a + 3b = 12 and 2a – 3b = 20
We have to find value of a.
2a + 3b = 12
+ 2a – 3b = 20
——————–
4a = 32
a = 8
39.) Simplify: ((x + y) 2 – (x – y) 2)/ (xy)
ANSWER:
((x + y) 2 – (x – y) 2)/ (xy)
We write (x + y) 2 as x2 + y2 + 2xy
And (x – y) 2 as x2 + y2– 2xy
((x + y) 2 – (x – y) 2)/ (xy) = x2 + y2 + 2xy – x2 – y2 + 2xy / (xy)
((x + y) 2 – (x – y) 2)/ (xy) = 4xy / xy
((x + y) 2 – (x – y) 2)/ (xy) = 4
40.) Find Dividend when Divisor = x + 3, Quotient = x + 1 and Remainder = 0
ANSWER:
We know,
Dividend = Divisor x Quotient + Remainder
Dividend = (x + 3) x (x + 1)
Dividend = x2 + 4x + 3
41.) Find Quotient if Dividend = y 2, Divisor = y – 5 and Remainder = 25
ANSWER:
We know,
Dividend = Divisor x Quotient + Remainder
Quotient = Dividend – Remainder / Divisor
Quotient =y 2 – 25 / y – 5
Quotient =y + 5
42.) If x – 1/x = 7 find x 2 + 1/(x 2)
ANSWER:
Given, x – 1/x = 7
We have to find x 2 + 1/(x 2)
We know,
(a – b) 2 = a2 + b2 – 2ab
a = x b = 1/x
(x – 1/x) 2 = x2 + 1/ (x2) – 2x x 1/x
(x – 1/x) 2= x2 + 1/ (x2) – 2
x2 + 1/ (x2) = (x – 1/x) 2 + 2
x2 + 1/ (x2) = 72 + 2
x2 + 1/ (x 2) = 51
43.) If z + 1/z = 11 find z 2 + 1/ (z 2)
ANSWER:
We know,
(a + b) 2 = a2 + b2 + 2ab
a = z b = 1/z
(z + 1/z) 2 = z2 + 1/ (z2) + 2z x 1/z
(z + 1/z) 2= z2 + 1/ (z2) + 2
z2 + 1/ (z2) = (z + 1/z) 2 – 2
z2 + 1/ (z2) = 112 – 2
z2 + 1/ (z2) = 119
44.) Express x 2 + 7x + 12 as product of two expressions.
ANSWER:
x2 + 7x + 12
From this we chose 2 numbers having product 12 and sum is 7.
3 and 4 = 3 x 4 = 12 and 3 + 4 = 7
x2 + 7x + 12
x(x + 3) + 4 (x + 3)
(x + 4) (x + 3) = x 2 + 7x + 12
45.) Simplify: (x 2 + y 2)/(x 4 – y 4)
ANSWER:
(x2 + y 2)/(x 4 – y 4)
We know,
(a2 – b2) = (a + b) (a – b)
x4 – y4 = (x2 – y2) (x2 + y2)
We put x4 – y4 = (x 2 – y 2) (x 2 + y 2)
(x2 + y 2)/(x 4 – y 4) = (x 2 + y 2)/ (x 2 – y 2) (x 2 + y 2)
(x2 + y 2)/(x 4 – y 4) = 1/(x 2 – y 2)
46.) Evaluate: (9.5(3 + 1.5))/ (1 + 3.5)
ANSWER:
(9.5(3 + 1.5))/ (1 + 3.5) = 9.5 x 4.5 / 4.5
(9.5(3 + 1.5))/ (1 + 3.5) = 9.5
47.) Evaluate: (7.4) 2 – (2.6) 2
ANSWER:
(7.4) 2 – (2.6) 2
This is in the form of (a2 – b2)
We know,
(a2 – b2) = (a + b) (a – b)
(7.4) 2 – (2.6) 2 = (7.4 + 2.6) (7.4 – 2.6)
(7.4) 2 – (2.6) 2 = 10 x 4.8
(7.4) 2 – (2.6) 2 = 48
48.) What will be the value of y 3 + y 2 – y + 1 if y = 1?
ANSWER:
y3 + y 2 – y + 1
Value of y = 1
13 + 12 – 1 + 1
= 1 + 1 – 1 + 1
y3 + y 2 – y + 1 = 2
49.) What will be the value of z 3 – z 2 + z + 2 if z = – 1?
ANSWER:
z3 – z 2 + z + 2
Value ofz = – 1
-13 – (-1)2 + (-1) + 2
-1 -1 -1 + 2
-3 + 2
z3 – z 2 + z + 2 = -1
50.) Evaluate: (7.2 x 2.8)/ (10 – 2.8)
ANSWER:
(7.2 x 2.8)/ (10 – 2.8)
We know,
(10 – 2.8) = 7.2
(7.2 x 2.8)/7.2
(7.2 x 2.8)/ (10 – 2.8) = 2.8