## MP Board Class 8th Maths Solutions Chapter 14 गुणनखंडन Ex 14.2

प्रश्न 1.

निम्नलिखित व्यंजकों के गुणनखण्ड कीजिए –

- a
^{2}+ 8a + 16 - p
^{2}– 10p + 25 - 25m
^{2}+ 30m + 9 - 49y
^{2}+ 84yz + 36z^{2} - 4x
^{2}– 8x + 4 - 121b
^{2}– 88bc + 16c^{2} - (l + m)
^{2}– 4lm - a
^{4}+ 2a^{2}b^{2}+ b^{4}

हल:

1. a^{2} + 8a + 16 = (a)^{2} + 2 x a x 4 + (4)^{2}

[∴ a^{2} + 2ab + b^{2} = (a + b)^{2}]

= (a + 4)^{2}

2. p^{2} – 10p + 25 = (p)^{2} – 2 x p x 5 + (5)^{2}

[∴ a^{2} – 2ab + b^{2} = (a – b)]

= (p – 5)^{2}

3. 25m^{2} + 30m + 9 = (5m)^{2} + 2 x 5m x 3 + (3)^{2}

= (5m + 3)^{2}

4. 49y^{2} + 84yz + 36z^{2}

= (7y)^{2} + 2 x 7y x 6z + (6z)^{2}

= (7y + 6z)^{2}

5. 4x^{2} – 8x + 4 = (2x)^{2} – 2 x 4x × 2 + (2)^{2}

= (2x – 2)^{2}

6. 121b^{2} – 88bc + 16c^{2}

= (11b)^{2} – 2 x 11b x 4c + (4c)^{2}

= (11b – 4c)

7. (l + m)^{2} – 4lm = l^{2} + 2lm + m^{2} – 4lm

= l^{2} – 2lm + m^{2}

= (l)^{2} – 2 x 1 x m + (m)^{2} = (1 – m)

8. a^{4} + 2a^{2}b^{2} + b^{4} = (a^{2})^{2} + 2 x a^{2} x b^{2} + (a)^{2}

= (a + b)^{2}

प्रश्न 2.

गुणनखण्ड कीजिए –

- 4p
^{2}– 9q^{2} - 63a
^{2}– 112b^{2} - 49x
^{2}– 36 - 16x
^{5}– 144x^{3} - (l + m)
^{2}– (l – m) - 9x
^{2}y^{2}– 16 - (x
^{2}– 2xy +y^{2}) – z^{2} - 25a
^{2}– 4b^{2}+ 28bc – 49c^{2}

हल:

1. 4p^{2} – 9q^{2}

a^{2} – b^{2} = (a – b) (a + b)

4p^{2} – 9q^{2} = (2p)^{2} – (3q)^{2}

= (2p – 3q) (2p + 3q)

2. 63a^{2} – 112b^{2} = 7 (9a^{2} – 16b^{2})

= 7 {(3a)^{2} – (4b)^{2}}

= 7 (3a – 4b) (3a + 4b)

3. 49x^{2} – 36 = (7x)^{2} – (6)^{2}

= (7x – 6) (7x + 6)

4. 16x^{5} – 144x^{3} = 16x^{3} (x^{2} – 9)

= 16x^{3} (x^{2} – 3^{2})

= 16x^{3} (x – 3) (x + 3)

5. (l + m)^{2} – (l – m) = [(l + m) – (l – m)][(l + m) – (l – m)]

= (l + m – 1 + m) (l + m + l – m)

= 2m x 2l = 4lm

6. 9x^{2}y^{2} – 16 = (3xy)^{2} – (4)^{2}

= (3xy – 4) (3xy + 4)

7. x^{2} – 2xy + y^{2} – z^{2} = (x – y)^{2} – z^{2}

= [(x – y) – z] [(x – y) + z]

= (x – y – z) (x – y + z)

8. 25a^{2} – 4b^{2} + 28bc – 49c^{2}

= 25a^{2} – (4b^{2} – 28bc + 49c^{2})

= 25a^{2} – [(2b)^{2} – 2 x 26 x 7c + (7c)^{2}]

= (5a)^{2} – (2b – 7c)^{2}

= [5a – (2b – 7c)] [5a + (2b – 7c)]

= (5a – 25 + 7c) (5a + 2b – 7c)

प्रश्न 3.

निम्नलिखित व्यंजकों के गुणनखण्ड कीजिए –

- ax
^{2}+ bx - 7p
^{2}+ 21q^{2} - 2x
^{2}+ 2xy^{2}+ 2xz^{2} - am
^{2}+ bm^{2}+ bn^{2}+ an^{2} - (lm + 1) + m + 1
- y (y + z) + 9 (y + z)
- 5y
^{2}– 20y – 8z + 2yz - 10ab + 4a + 5b + 2
- 6xy – 4y + 6 – 9x.

हल:

1. ax^{2} + bx = x (ax + b)

2. 7p^{2} + 21q^{2} = 7 (p^{2} + 3q^{2})

3. 2x^{2} + 2xy^{2} + 2xz^{2} = 2x (x^{2} + y^{2} + z^{2})

4. amv + bm^{2} + bn^{2} + an^{2}

= (am^{2} + bm^{2}) + (bn^{2} + an^{2})

= m^{2} (a + b) + n^{2} (b+ a)

= (a + b) (m^{2} + n^{2})

5. (lm + 1) + m + 1 = 1(m + 1) + 1 (m + 1)

= (m + 1) (1 + 1)

6. y (y + z) + 9 (y + z) = (y + z) (y + 9)

7. 5y^{2} – 20y – 8z + 2yz

= (5y^{2} – 20y) + (2yz – 8z)

= 5y (y – 4) + 2z (y – 4).

= (y – 4) (5y + 2z)

8. 10ab + 4a + 56+2

= (10ab + 5b) + (4a + 2)

= 5b (2a + 1) + 2 (2a + 1)

= (2a + 1) (5b + 2)

9. 6xy – 4y + 6 – 9x

= (6xy – 4y) – (9x – 6)

= 2y (3x – 2) – 3 (3x – 2)

= (3x – 2) (2y – 3)

प्रश्न 4.

गुणनखण्ड कीजिए –

- a
^{4}– b^{4} - p
^{4}– 81 - x
^{4}(y + z)^{4} - x
^{4}– (x – z)^{4} - a
^{2}– 2a^{2}b^{2}+ b^{4}

हल:

1. a^{4} – b^{4} = (a^{2})^{2} – (b^{2})^{2}

= (a^{2} + b^{2}) (a^{2} – b^{2})

= (a^{2} + b^{2}) (a + b) (a – b)

2. p^{4} – 81 = (p^{2})^{2} – (9)^{2}

= (p^{2} + 9) (p^{2} – 9)

= (p^{2} + 9) (p + 3) (p – 3)

3. x^{4} – (y + z)^{4} = (x^{2})^{2} – [(y + 2)^{2}]^{2}

= [x^{2} – (y + z)^{2}] [x^{2} + (y + z)^{2}]

= [x – (y + z)] [x + (y + z)] [x^{2} + (y + z)^{2}]

= (x – y – z) (x + y + z) [x^{2} + (y + z)^{2}]

4. x^{4} – (x – z)^{4} = (x^{2})^{2} – [(x – z)^{2}]^{2}

= [x^{2} – (x – z)^{2}] [x^{2} + (x – z)]

= [x – (x – z)] [x + (x – z)] [x^{2} + (x^{2} – 2x^{2} + z^{2})] = (x – x + z) (x + x – z)(2x^{2} – 2xz + z^{2})

= z(2x – z) (2x^{2} – 2xz + z^{2})

5. a^{4} – 2a^{2}b^{2} + b^{2} = (a^{2})^{2} – 2 x a^{2} x b^{2} + (b^{2})^{2}

= [a^{2} – b^{2}]^{2}

= [(a – b) (a + b)]^{2}

= (a – b)^{2} (a + b)^{2}

प्रश्न 5.

निम्नलिखित व्यंजकों के गुणनखण्ड कीजिए –

- p
^{2}+ 6p + 8 - q
^{2}– 10q + 21 - p
^{2}+ 6p – 16

हल:

1. P^{2} + 6p + 8 = p^{2} + (4 + 2) p + 8

(∴8 = 4 x 2)

= p^{2} + 4p + 2p +8

= p (p + 4) + 2 (p + 4)

= (p + 4) (p + 2)

2. q^{2} – 10q + 21 = q^{2} – (7 + 3) q + 21

(∴ 21 = 3 x 7)

=q^{2} – 7q – 3q + 21

= q(q – 7) – 3 (q – 7)

= q (q – 7) (q – 3)

= (q – 3) (q – 7)

3. p^{2} + 6p – 16 = p^{2} + (8 – 2)p – 16

(∴ 16 = 8 x 2)

= p^{2} + 8p – 2p – 16

= p (p + 8) – 2 (p + 8)

= (p + 8) (p – 2)

पाठ्य-पुस्तक पृष्ठ संख्या # 234

प्रयास कीजिए (क्रमांक 14.2)

प्रश्न 1.

भाग दीजिए –

- 24xy
^{2}3z^{3}को 6yz^{2}से - 63a
^{2}b^{4}c^{6}को 7a^{2}b^{2}c^{3}से।

हल:

1. 24xy^{2}z^{3} + 6yz^{2}

= 4xyz

2. 63a^{2}b^{4}c^{6} ÷ 7a^{2}b^{2}c^{3}

= 9b^{2}c^{3}