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The method of writing a given expression as the product of two or some more factors is called factorisation. The product of the greatest common factors of the numerical coefficients and the common letters with the smallest powers is the greatest common factor of two or more monomials. For example, we know the factors of expressions like 5ab, 5a2b, (x + 2) (x + 3), and 5x (6y+2). But to find the factors for expressions like 6xy − 4y + 6 − 9x, 4y2 − 12y + 9, and 15x2y + 24xy2, we need to use the method of common factors.
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1. The factorisation of 12a2b + 15ab2 gives
A. 3ab(4ab + 5)
B. 3ab(4a + 5b)
C. 3a(4a + 5b)
D. 3b (4a + 5b)
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Answer: B. 3ab(4a + 5b)
Explanation:
12a2b + 15ab2
12a2b = 3 × 4 × a × a × b
15ab2 = 3 × 5 × a × b × b
The common factors are 3ab.
12a2b + 15ab2 = 3ab(4a + 5b)
2. The factorisation of 12x + 36 is
A. 12(x + 3)
B. 12(3x)
C. 12(3x + 1)
D. x(12 + 36x)
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Answer: A. 12(x + 3)
Explanation:
12x + 36
12x + 12 (3)
=12(x + 3)
3. On factorising 14pq + 35pqr, we get
A. pq(14 + 35r)
B. p(14q + 35qr)
C. q(14p + 35pr)
D. 7pq(2 + 5r)
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Answer: D. 7pq(2 + 5r)
Explanation:
14pq + 35pqr
= (2) (7) (p) (q) + (5) (7) (p) (q) (r)
= 7pq(2 + 5r)
4. The factors of 6xy − 4y + 6 − 9x are
A. (3x + 2) (2y + 3)
B. (3x − 2) (2y − 3)
C. (3x − 2) (2y + 3)
D. (3x − +2) (2y − 3)
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Answer: B. (3x − 2) (2y − 3)
Explanation:
6xy − 4y + 6 − 9x
= 6xy − 4y − 9x + 6
= 2y (3x − 2) − 3 (3x − 2)
= (3x − 2) (2y − 3)
5. The factors of x2 + xy + 8x + 8y are
A. (x + y) (x + 8)
B. (2x + y) (x + 8)
C. (x + 2y) (x + 8)
D. (x + y) (2x + 8)
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Answer: A. (x + y) (x + 8)
Explanation:
x2 + xy + 8x + 8y
= x(x + y) + 8(x + y)
= (x + y) (x + 8)
6. The factors of 4y2 − 12y + 9 are
A. (2y + 3)2
B. (2y − 3)2
C. (2y − 3) (2y + 3)
D. None of the above
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Answer: B. (2y − 3)2
Explanation:
4y2 − 12y + 9
4y2 = (2y)2 & 9 = 32 & 12y = 2 × 3 × 2y
4y2 − 12y + 9 = (2y)2 − 2 × 3 × (2y) + (3)2
= (2y − 3)2 [By algebraic identities: (a − b)2 = a2 + b2 − 2ab]
7. The factors of 49p2 − 36 are
A. (7p + 6)2
B. (7p−6)2
C. (7p − 6) (7p + 6)
D. None of the above
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Answer: C. (7p − 6) (7p + 6)
Explanation:
49p2 − 36
= (7p)2 − (6 )2
= (7p − 6) (7p + 6)
8. The factors of m2 − 256 are
A. (m + 4)2
B. (m − 4)2
C. (m − 4) (m + 4)
D. None of the above
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Answer: D. None of the above
Explanation:
m2 = (m)2 and 256 = (16)2
By using the algebraic identity,
a2 − b2 = (a + b) (a−b),
we get (m + 16) (m−16).
9. When we factorise x2 + 5x + 6, then we get
A. (x + 2) (x + 3)
B. (x − 2) (x − 3)
C. (x × 2) + (x × 3)
D. (x × 2) − (x × 3)
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Answer: A. (x + 2) (x + 3)
Explanation:
The factors of a form:
(x + a) (x + b) = x2 + (a + b) x + ab
x2 + 5x + 6
a + b = 5 and ab = 6
x2 + 5x + 6 = (x + 2) (x + 3)
10. The factors of 3m2 + 9m + 6 are
A. (m + 1) (m + 2)
B. 3(m + 1) (m + 2)
C. 6(m + 1) (m + 2)
D. 9(m + 1) (m + 2)
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Answer: B. 3(m + 1) (m + 2)
Explanation:
3m2 + 9m + 6 = 3(m2 + 3m + 2)
= 3 [m2 + m + 2m + 2]
= 3 [m (m + 1) + 2(m + 1)]
= 3 [(m + 1) (m + 2)]
11. The common factor of a3b3 and ab2 is
A. a2b2
B. ab2
C. a2b
D. ab
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Answer: B. ab2
Explanation:
a3b3 = a × a × a × b × b × b
ab2 = a × b × b
The common factors are a × b × b = ab2
12. The common factor 12a and 30 is
A. 6
B. 12
C. 30
D. 6a
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Answer: A. 6
Explanation:
12a = 2 × 2 × 3 × a
30 = 2 × 3 × 5
13. The common factors of 10a, 20b, and 30c are
A. ab
B. ac
C. 10abc
D. 10
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Answer: D. 10
Explanation:
10a = 2 × 5 × a
20b = 2 × 2 × 5 × b
30c = 2 × 3 × 5 × c
The common factor is 10.
14. The common factor of 24a3b4, 36a4c4, and 48a3b2c is
A. 12a3
B. 24a3
C. 36a3
D. 48a3
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Answer: A. 12a3
Explanation:
24a³b4 = 2 × 2 × 2 × 3 × a × a × a × b × b × b × b
36a4c4 = 2 × 2 × 3 × 3 × a × a × a × a × c × c × c × c
48a3b2c = 2 × 2 × 2 × 2 × 3 × a × a × a × b × b × c
The common factor is 12a3
15. The common factor of 8a2b4c2, 12a4bc4 and 20a3b4 is
A. a4b4
B. a2b2
C. 4a2b2
D. 4a2b
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Answer: D. 4a2b
Explanation:
8a2b4c2 = 2 × 2 × 2 × a × a × b × b × b × b × c × c
12a4bc2 = 2 × 2 × 3 × a × a × a × a × b × c × c
20a3b4 = 2 × 2 × 5 × a × a × a × b × b × b × b
The common factor is 4a2b.
16. The common factor of 6a3b4c2, 21a2b, and 15a3 is
A 3a2
B 3a3
C 6a3
D 6a2
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Answer: A. 3a2
Explanation:
6a3b4c2 = 2 × 3 × a × a × a × b × b × b × b × c × c
21a2b = 3 × 7 × a × a × a
15a364c4 = 3 × 5 × a × a × a
The common factor is 3a2.
17. The common factor of 2a2b4c2, 8a4b3c4, and 6a3b4c2 is
A 2a2b3c2
B 6a2b3c2
C 8a2b3c2
D a4b4c4
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Answer: A. 2a2b3c2
Explanation:
6a2b4c2 = 2 × a × a × b × b × b × b × c × c × c × c
8a4b3c4 = 2 × 2 × 2 × a × a × a × a × b × b × b × c × c × c × c
6a3b4c2 = 2 × 3 × a × a × a × b × b × b × b × c × c.
The common factor is 2a2b3c2.
18. What is the common factor of 3a2b4c2, 12b2c4, and 15a3b4c4?
A b4c4
B 3b2c2
C 15b2c4
D 12b2c4
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Answer: B. 3b2c2
Explanation:
3a²b4c² = 3 × a × a × b × b × b × b × c × c
12b²c4 = 2 × 2 × 3 × b × b × c × c × c × c.
15a³b4c4 = 3 × 5 × a × a × a × b × b × b × b × c × c × c × c.
The common factor is 3b2c2.
19. The common factor of 24x3y4, 36x4z4, and 48x3y2z is
A 12x3
B 24x3
C 36x3
D 48x3
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Answer: A. 12x3
Explanation:
24x³y4 = 2 × 2 × 2 × 3 × x × x × x × y × y × y × y.
36x4z4 = 2 × 2 × 3 × 3 × x × x × x × x × z × z × z × z.
48x3y2z = 2 × 2 × 2 × 2 × 3 × x × x × x × y × y × z
The common factor is 12x3.
20. What is the common factor of 72x3y4z4, 120z2d4x4, and 96y3z4d4 ?
A 96z3
B 120z3
C 72z3
D 24z2
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Answer: D. 24z2
Explanation:
72x3y4z4 = 2 × 2 × 2 × 3 × 3 × x × x × x × y × y × y × y × z × z × z × z.
120z²d4x4 = 2 × 2 × 2 × 3 × 5 × z × z × d × d × d × d × x × x × x × x
96y3z4d4 = 2 × 2 × 2 × 2 × 2 × 3 × y × y × z × z × z × z × d × d × d × d
The common factor is 24z2.
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