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x²(a - b) - y²(a - b) + z²(a - b)

= (a - b) (x² - y² + z²)

(i) x² + xy + 8x + 8y 

= x × x + x × y + 8 × x + 8 × y

= x (x + y) + 8 (x + y)

= (x + y) (x + 8)

(ii) 15xy − 6x + 5y − 2 

= 3 × 5 × x × y − 3 × 2 × x + 5 × y − 2

= 3x (5y − 2) + 1 (5y − 2)

= (5y − 2) (3x + 1)

(iii) ax + bx − ay − by 

= a × x + b × x − a × y − b × y

= x (a + b) − y (a + b)

= (a + b) (x − y)

(i) 15pq + 15 + 9q + 25p 

= 15pq + 9q + 25p + 15

= 3 × 5 × p × q + 3 × 3 × q + 5 × 5 × p + 3 × 5

= 3q (5p + 3) + 5 (5p + 3)

= (5p + 3) (3q + 5)

(ii) z − 7 + 7xy − xyz 

= z − x × y × z − 7 + 7 × x × y

= z (1 − xy) − 7 (1 − xy)

= (1 − xy) (z − 7)

(i) x² + xy + 8x + 8y 

= x × x + x × y + 8 × x + 8 × y

= x (x + y) + 8 (x + y)

= (x + y) (x + 8)

(ii) 15xy − 6x + 5y − 2 

= 3 × 5 × x × y − 3 × 2 × x + 5 × y − 2

= 3x (5y − 2) + 1 (5y − 2)

= (5y − 2) (3x + 1)

(iii) ax + bx − ay − by 

= a × x + b × x − a × y − b × y

= x (a + b) − y (a + b)

= (a + b) (x − y)

We have,

12x² – 17xy + 6y²

The coefficient of x² is 12

The coefficient of x is -17y

Constant term is 6y²

So, we express the middle term -17xy as -9xy – 8xy

12x² -17xy+ 6y² = 12x² – 9xy – 8xy + 6y²

= 3x (4x – 3y) – 2y (4x – 3y)

= (3x – 2y) (4x – 3y)

We have,

14x² + 11xy – 15y²

The coefficient of x² is 14

The coefficient of x is 11y

Constant term is -15y²

So, we express the middle term 11xy as 21xy – 10xy

14x² + 11xy- 15y² = 14x² + 21xy – 10xy – 15y²

= 2x (7x – 5y) + 3y (7x – 5y)

= (2x + 3y) (7x – 5y)

We have,

3x² + 22x + 35

The coefficient of x² is 3

The coefficient of x is 22

Constant term is 35

So, we express the middle term 22x as 15x + 7x

3x² + 22x + 35 = 3x² + 15x + 7x + 35

= 3x (x + 5) + 7 (x + 5)

= (3x + 7) (x+ 5)

We have,

7x – 6x² + 20

– 6x² + 7x + 20

6x² – 7x – 20

The coefficient of x² is 6

The coefficient of x is -7

Constant term is -20

So, we express the middle term -7x as -15x + 8x

6x² – 7x – 20 = 6x² – 15x + 8x – 20

= 3x (2x – 5) + 4 (2x – 5)

= (3x + 4) (2x – 5)

We have,

11x² – 54x + 63

The coefficient of x² is 11

The coefficient of x is -54

Constant term is 63

So, we express the middle term -54x as -33x – 21x

11x² – 54x + 63 = 11x² – 33x – 21x – 6

= 11x (x – 3) – 21 (x – 3)

= (11x – 21) (x – 3)

By taking 2a-b as a common factor for the above equation we get,

x³ (2a-b) + x²(2a-b) = (2a-b) (x³ + x²)

x(a-3) –y(a-3)

By taking (a-3) as a common factor for the above equation we get,

x(a-3) –y(a-3) = (a-3) (x-y)

By taking x+3 as a common factor for the above equation we get,

x (x+3) + 5(x+3) = (x+3) (x + 5)

let’s take HCF of above equation By taking 5 as a common factor for the above equation we get,

-5 -10t + 20t² = -5 (1 +2t -4t²)

3ac + 7bc – 3ad – 7bd 

c(3a + 7b)-d(3a + 7b) (3a + 7b) (c-d)

3ab + 3a + 2b + 2 

= [3 × a × b + 3 × a] + [2 × b + 2] 

= 3 × a [b + 1] + 2 [b + 1] 

= (b + 1) (3a + 2)

(3a + 4b)(a – b) = 3a² – 4a² 

3a(a – b) + 4b(a – b) = 3a² – 4a² 

3a² – 3ab + 4ab – 4b² = – a² 

3a² + ab – 4b² ≠ a²   

∴ The given sentence is wrong. 

Correct sentence is (3a + 4b) (a – b) = 3a² + ab – 4b²

(3x)² + 4x +7 = 3x² + 4x +7 

⇒ (3x)² = 3x² 

⇒ 9x² = 3x² 

⇒ 9 = 3 It is not possible 

∴ The given sentence is wrong. 

Correct sentence is (3x)² + 4x + 7 = 9x² + 4x + 7.

(2x)² + 5x = 4x + 5x = 9x 

⇒ 4x² + 5x = 4x + 5x 

⇒ 4x² = 4x

⇒ x² = x 

⇒ x ≠ √x 

∴ The given sentence is wrong. 

Correct sentence is (2x)² + 5x = 4x² + 5x.

(2a + 3)² = 2a² + 6a +9

⇒ (2a)² + 2 × 2a × 3 + 3² = 2a² + 6a + 9

⇒ 4a² + 12a + 9 = 2a²+ 6a + 9

⇒ 4a² – 2a² = 6a – 12a

⇒ 2a² = – 6a

⇒ 2a ≠ 6

∴ The given sentence is wrong.

Correct sentence is (2a + 3)² = 4a² + 12a + 9.

3(x – 9) = 3x – 9 

3(x – 9) = 3x – 9

⇒ 3x – 3 x 9 = 3x – 9 

⇒ 3x – 27 = 3x – 9 

⇒ – 27 ≠ – 9 

∴ The given sentence is wrong. 

Correct sentence is 3(x – 9) = 3x – 27.

2x³ + 1 ÷ 2x³ = 1

⇒  \(\frac{2x^3\,+\,1}{2x^3}\) = 1

In the denominator the term T is missing. 

∴ The given sentence is wrong. 

Correct sentence is 2x³ + 1 ÷ 2x³ = 1 + \(\frac{1}{2x^3}\)

3x + 2 ÷ 3x = \(\frac{2}{3x}\)

⇒ \(\frac{3x\,+\,2}{3x}\) = \(\frac{2}{3x}\)

⇒ 1 + \(\frac{2}{3x}\) = \(\frac{2}{3x}\)

⇒ 1 ≠ 0 

∴ The given sentence is wrong. 

Correct sentence is 3x + 2 ÷ 3x = 1 + \(\frac{2}{3x}\)

4p + 3p + 2p + p – 9p = 0 

⇒ 10p – 9p = 0 

⇒ p = 0 It is not possible 

∴ The given sentence is wrong.

Correct sentence is 4p + 3p + 2p + p – 9p – p = 0

4c² + 3c - 10 = 4c² + 8c - 5c - 10

= 4c(c + 2) -5(c + 2)

= (c + 2) (4c - 5)

 2x² + xy - 6y² = 2x² + 4xy - 3xy - 6y² 

= 2x(x + 2y) -3y(x + 2y)

= (x + 2y) (2x-3y)

pq = 32 and p + q = – 12

p = -8, q = – 4

A) 121 – (x – 3)² 

Correct option is (A) 121 – (x – 3)²

(8 + x) (14 – x) \(=112-8x+14x-x^2\)

\(=112+6x-x^2\)

\(=112+9-9+6x-x^2\)

\(=121-(9-6x+x^2)\)

\(=121-(x^2-2\times x\times3+3^2)\)

\(=121-(x-3)^2\)

Correct option is  D) 5x²

Correct option is (D) 5x²

\(70\,x^4\div14\,x^2=\frac{70\,x^4}{14\,x^2}\) \(=5x^2\)

Correct option is  D) m – 16

Correct option is (D) m – 16

\((m^2–14\,m–32)\div(m+2)\) \(=\frac{m^2–14\,m–32}{m+2}\)

\(=\frac{m^2–16\,m+2\,m–32}{m+2}\)

\(=\frac{m(m-16)+2(m-16)}{m+2}\)

\(=\frac{(m-16)(m+2)}{m+2}\)

= m – 16

Correct option is  A) 6a

Correct option is (A) 6a

\((6a^2+30a)\div(a+5)=\frac{6a^2+30a}{a+5}\)

\(=\frac{6a(a+5)}{a+5}\)

= 6a

(i) 12x, 36

12x = 2 × 2 × 3 × x

36 = 2 × 2 × 3 × 3

∴ Common factor = 2 × 2 × 3 = 12

(ii) 14pq, 28p²q²

14pq = 2 × 7 × p × q

28p²q² = 2 × 2 × 1 × p × p × q × q

∴ Common factor = 2 × 7 × p × q = 14pq

(iii) 6abc, 24ab², 12a²b

6abc = 2 × 3 × a × b × c

24 ab² = 2 × 2 × 2 × 3 × a × b × b

12 a²b = 2 × 2 × 3 × a × a × b

∴ Common factor = 2 × 3 × a × b = 6ab

(iv) 16x³, – 4x², 32x

16x³ = 2 × 2 ×2 × 2 × x × x × x

– 4x² = (- 1) × 2 × 2 × x × x

32x = 2 × 2 × 2 × 2 × 2 × x

∴ Common factor = 2 × 2 × x = 4x

(v) 10pq, 20qr, 30rp

10pq = 2 × 5 × p × q

20qr = 2 × 2 × 5 × q × r

30rp = 2 × 3 × 5 × r × p

∴ Common factor = 2 × 5 = 10

(vi) 3x²y³, 10x²y², 6x²y²z

3x²y³ = 3 × x × x × y × y × y

10x²y² = 2 × 5 × x × x × y × y

6x²y²z = 2 × 3 × x × x × y × y × z

∴ Common factor = x × x × y × y = x²y²

6a² - 3a²b - bc² + 2c² 

= 6a² - 3a²b + 2c² - bc² 

= 3a²(2-b) + c²(2-b)

= (2-b) (3a² + c²)

Correct option is  C) x + 6

Correct option is (C) x + 6

\(x(3x^2–108)\div3x(x–6)\) \(=\frac{x(3x^2–108)}{3x(x–6)}\)

\(=\frac{3x(x^2-36)}{3x(x–6)}\) \(=\frac{3x(x-6)(x+6)}{3x(x–6)}\)

= x + 6

B) 3x² + 5x + 4

Correct option is (B) 3x² + 5x + 4

\((6x^4+10x^3+8x^2)\div2x^2\) \(=\frac{6x^4+10x^3+8x^2}{2x^2}\)

\(=\frac{2x^2(3x^2+5x+4)}{2x^2}\)

= \(3x^2+5x+4\)

(i) 12x = 2 × 2 × 3 × x

36 = 2 × 2 × 3 × 3

(ii) 2y = 2 × y

22xy = 2 × 11 × x × y

(iii) 14pq = 2 × 7 × p × q

28p²q² = 2 × 2 × 7 × p × p × q × q

The common factors are 2, 7, p, q.

And, 2 × 7 × p × q = 14pq

A) 5 (a + b + c)

Correct option is (A) 5 (a + b + c)

\(30\,(a^2bc+ab^2c+abc^2)\div6abc\) \(=\frac{30\,(a^2bc+ab^2c+abc^2)}{6abc}\)

\(=\frac{30abc(a+b+c)}{6abc}\)

= 5 (a + b + c)

3a²b - 12a² - 9b + 36

= 3a²(b - 4) - 9(b - 4)

= (b - 4) (3a² - 9) = (b - 4) 3(a² - 3)

= 3(b - 4) (a² - 3)

Correct option is  C) 2, 4, 8

Correct option is (C) 2, 4, 8

Common factors of 8x & 24 are 2, 4 & 8.

D) (m – 7) (m + 3)

Correct option is (D) (m – 7) (m + 3)

\(m^2–4m–21\) \(=m^2-7m+3m-21\)

= m(m - 7) + 3(m - 7)

= (m – 7) (m + 3)

(i) 10pq = 2 × 5 × p × q

20qr = 2 × 2 × 5 × q × r

30rp = 2 × 3 × 5 × r × p

The common factors are 2, 5.

And, 2 × 5 = 10

(ii) 3x²y³ = 3 × x × x × y × y × y 

10x³y² = 2 × 5 × x × x × x × y × y 

6x²y²z = 2 × 3 × x × x × y × y × z

x × x × y × y = x²y²

(iii) 3(2m-3)(2m+3)

By taking 3 as the common factor we get,

By using the formula a² – b² = (a – b) (a + b)

(12m² -27) 

= 3(4m² -9)

= 3(2m -3) (2m + 3)

(i) 2x = 2 × x

3x² = 3 × x × x

4 = 2 × 2

The common factor is 1.

(ii) 6abc = 2 × 3 × a × b × c

24ab² = 2 × 2 × 2 × 3 × a × b × b

12a²b = 2 × 2 × 3 × a × a × b

The common factors are 2, 3, a, b.

And, 2 × 3 × a × b = 6ab

(iii) 16x³ = 2 × 2 × 2 × 2 × x × x × x −4x² = −1 × 2 × 2 × x × x

32x = 2 × 2 × 2 × 2 × 2 × x

The common factors are 2, 2, x.

And, 2 × 2 × x = 4x

i) 8x = 2 × 2 × 2 × x

24 = 8 × 3 = 2 × 2 × 2 × 3

∴ Common factors of 8x, 24 = 2, 4, 8.

ii) 3a, 21ab

3a = 3 × a

21ab = 7 × 3 × a × b

∴ Common factors of 3a, 21ab = 3, a, 3a.

iii) 7xy, 35x²y³

7xy = 7 × x × y

35x²y³ = 7 × 5 × x × x × y × y × y

∴ Common factors of 7xy, 35x²y³ = 7, x, y, 7x, 7y, xy, 7xy.

iv) 4m², 6m², 8m³

4m² = 2 × 2 × m × m

6m² = 2 × 3 × m × m

8m³ = 2 × 2 × 2 × m × m × m

∴ Common factors of 4m² , 6m² , 8m³ = 2, m, m², 2m, 2m².

(iii) (x+y)(x-z)

By taking x and y as the common factor we get,

x² –xz + xy –yz 

= x(x-z) + y(x-z)

= (x-z) (x+y)

First find the two numbers whose sum= 6 and product= -91

Clearly, the numbers are 13 and 7

∴ we get, a² + 6a – 91 

= a² + 13a – 7a – 91

= a (a+13) – 7(a+13)

= (a+13) (a-7)

(iv) 2x (1 – 4x) (1 + 4x)

let us consider (2x – 32x³) = 2x (1 – 16x²) by taking 2x as the common factor

By using the formula a² – b² = (a – b) (a + b) 

= 2x (1 – 4x) (1 + 4x)

(iv) 7(a – 3b) (a + 3b)

let us consider (7a² – 63b²) = 7(a² – 9b²) by taking 7 as the common factor

By using the formula a² – b² = (a – b) (a + b) 

= 7 (a – 3b) (a + 3b)

L.H.S. = 3/(4x + 3) ≠ R.H.S

The correct statement is  3/(4x + 3) =  3/(4x + 3)

3x⁵ – 48x³ can be written as 3x³(x² – 16)

By using the formula a² – b² = (a+b) (a-b)

Now solving for the above equation

3x³(x² – 16) 

= 3x³((x)² – (4)²)

= 3x³(x+4) (x-4)