**1. Factorise the following expressions.**

**(i) a ^{2}+8a+16**

**Solution:**

Here we observe that 4 + 4 = 8 and 4 × 4 = 16

Now, we split 8a

= a^{2} + 4a + 4a + 16

= (a^{2} + 4a) + (4a + 16)

= a (a + 4) + 4(a + 4)

= (a + 4) (a + 4)

= (a + 4)^{2}

**(ii) p ^{2}–10p+25**

**Solution:**

Here we observe that, 5 + 5 = 10 and 5 × 5 = 25

Now, we split 10 p

= p^{2} – 5p – 5p + 25

= (p^{2} – 5p) + (-5p + 25)

= p (p – 5) – 5(p – 5)

= (p – 5) (p – 5)

= (p – 5)^{2}

= p^{2}-2×5×p+5^{2}

= (p-5)^{2}

**(iii) 25m ^{2}+30m+9**

**Solution:**

Here we observe that, 15 + 15 = 30 and 15 × 15 = 25 × 9 = 225

Now, we split 30m

= 25m^{2} + 15m + 15m + 9

= (25m^{2} + 15m) + (15m + 9)

= 5m (5m + 3) + 3(5m + 3)

= (5m + 3) (5m + 3)

= (5m + 3)^{2}

**(iv) 49y ^{2}+84yz+36z^{2}**

**Solution:**

Here we observe that, 42 + 42 = 84 and 42 × 42 = 49 × 36 = 1764

Now, we split 84yz

= 49y^{2} + 42yz + 42yz + 36z^{2}

= 7y (7y + 6z) +6z (7y + 6z)

= (7y + 6z) (7y + 6z)

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

**(v) 4x ^{2}–8x+4**

**Solution:**

We can take 4 common in given expression

Now,

= 4(x^{2} – 2x + 1)

We split 2x, such that 1 x 1 = 1 and 1 + 1 = 2

= 4 (x^{2} – x – x + 1)

= 4 [x (x – 1) -1(x – 1)]

= 4 (x – 1) (x – 1)

= 4 (x – 1)^{2}

**(vi) 121b ^{2}-88bc+16c^{2}**

**Solution:**

Here we observe that, 44 + 44 = 88 and 44 × 44 = 121 × 16 = 1936

Now, we split 88bc

= 121b^{2} – 44bc – 44bc + 16c^{2}

= 11b (11b – 4c) – 4c (11b – 4c)

= (11b – 4c) (11b – 4c)

= (11b – 4c)^{2}

**(vii) (l+m) ^{2}-4lm (Hint: Expand (l+m)^{2 }first)**

**Solution:**

[ Using property (a + b)^{2} = a^{2} + 2ab + b^{2}]

After expanding (l + m)^{2},

l^{2} + 2lm + m^{2} – 4lm

= l^{2} – 2lm + m^{2}

Now, we split 2lm

= l^{2 }– Im – lm + m^{2}

= l(l – m) – m(l – m)

= (l – m) (l – m)

= (l – m)^{2}

**(viii) a ^{4}+2a^{2}b^{2}+b^{4}**

**Solution:**

We split 2a^{2}b^{2}

= a^{4} + a^{2}b^{2} + a^{2}b^{2} + b^{4}

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

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

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

**2. Factorise.**

**(i) 4p ^{2}–9q^{2}**

**Solution:**

Using identity: a^{2}-b^{2} = (a + b) (a – b)

= (2p)^{2}-(3q)^{2}

= (2p-3q) (2p+3q)

**(ii) 63a ^{2}–112b^{2}**

**Solution:**

We take 7 as common

= 7(9a^{2} –16b^{2})

Using identity: a^{2} – b^{2} = (a + b) (a – b)

= 7((3a)^{2}–(4b)^{2})

= 7(3a+4b) (3a-4b)

**(iii) 49x ^{2}–36**

**Solution:**

Using identity: a^{2} – b^{2} = (a + b) (a – b)

= (7x)^{2} -6^{2}

= (7x+6) (7x–6)

**(iv) 16x ^{5}–144x^{3}**

**Solution:**

We take 16x^{3} as common

= 16x^{3}(x^{2}–9)

Using identity: a^{2} – b^{2} = (a + b) (a – b)

= 16x^{3} (x^{2} – 3^{2})

= 16x^{3}(x–3) (x+3)

**(v) (l+m) ^{2}-(l-m)^{2}**

**Solution:**

Using identity: a^{2} – b^{2} = (a + b) (a – b)

= {(l+m)-(l–m)} {(l +m) + (l–m)}

Now, simplify this equation

= (l+m–l+m) (l+m+l–m)

= (2m) (2l)

= 4 ml

**(vi) 9x ^{2}y^{2}–16**

**Solution:**

Using identity: a^{2} – b^{2} = (a + b) (a – b)

= (3xy)^{2}-4^{2}

= (3xy–4) (3xy+4)

**(vii) (x ^{2}–2xy+y^{2})–z^{2}**

**Solution:**

Using identity: a^{2} – b^{2} = (a + b) (a – b)

We can write (x^{2} – 2xy + y^{2}) as (x-y)^{2}

Now,

= (x–y)^{2}–z^{2}

Again, using identity: a^{2} – b^{2} = (a + b) (a – b)

= {(x–y)–z} {(x–y)+z}

= (x–y–z) (x–y+z)

**(viii) 25a ^{2}–4b^{2}+28bc–49c^{2}**

**Solution:**

= 25a^{2} – (4b^{2} – 28bc + 49c^{2})

= (5a)^{2} – (2b – 7c)^{2}

Using identity: a^{2} – b^{2} = (a + b) (a – b)

= {5a – (2b – 7c)} {5a + (2b – 7c)}

= (5a – 2b + 7c) (5a + 2b – 7c)

**3. Factorise the expressions.**

**(i)ax ^{2}+bx**

**Solution:**

Taking x as common:

= x(ax+b)

**(ii) 7p ^{2}+21q^{2}**

**Solution:**

Taking 7 as common:

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

**(iii) 2x ^{3}+2xy^{2}+2xz^{2}**

**Solution:**

Taking 2x as common:

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

**(iv) am ^{2}+bm^{2}+bn^{2}+an^{2}**

**Solution:**

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

Taking m^{2 }and n^{2} as common:

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

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

**(v) (lm+l)+m+1**

**Solution:**

= (lm+m) + (l+1)

Taking m as common:

= m(l+1) + (l+1)

= (m+1) (l+1)

**(vi) y (y + z) + 9(y + z)**

**Solution:**

= (y + 9) (y + z)

**(vii) 5y ^{2}–20y–8z+2yz**

**Solution:**

= (5y^{2}–20y) – (8z+2yz)

= 5y(y–4) +2z(y–4)

= (y–4) (5y+2z)

**(viii) 10ab+4a+5b+2**

**Solution:**

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

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

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

**(ix) 6xy–4y+6–9x**

**Solution:**

= (6xy–9x) – (4y+6)

= 3x(2y–3)–2(2y–3)

= (2y–3) (3x–2)

**4.Factorise.**

**(i) a ^{4}–b^{4}**

**Solution:**

We can write it as:

= (a^{2})^{2}-(b^{2})^{2}

Now, using identity: [a^{2} – b^{2} = (a + b) (a – b)]

= (a^{2}-b^{2}) (a^{2}+b^{2})

= (a – b) (a + b) (a^{2}+b^{2})

**(ii) p ^{4}–81**

**Solution:**

We can write it as:

= (p^{2})^{2}-(9)^{2}

Now, using identity: [a^{2} – b^{2} = (a + b) (a – b)]

= (p^{2}-9) (p^{2}+9)

We can write (p^{2} – 9 = p^{2} – 3^{2})

= (p^{2}-3^{2}) (p^{2}+9)

=(p-3) (p+3) (p^{2}+9)

**(iii) x ^{4}–(y+z) ^{4}**

**Solution:**

We can write this equation as:

= (x^{2})^{2}– [(y+z)^{2}]^{2}

Now, using identity: [a^{2} – b^{2} = (a + b) (a – b)]

= {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}}

**(iv) x ^{4}–(x–z) ^{4}**

**Solution:**

We can write as:

= (x^{2})^{2}-{(x-z)^{2}}^{2}

Now, using identity: [a^{2} – b^{2} = (a + b) (a – b)]

= (x^{2})^{2} – [(x – z)^{2}]^{2}

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

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

= (z)(2x-z) [(x^{2} + (x – z)^{2}]

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

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

(2xz – z^{2}) (2x^{2} -2xz + z^{2})

**(v) a ^{4}–2a^{2}b^{2}+b^{4}**

**Solution:**

Using identity: [a^{2} – b^{2} = (a + b) (a – b)]

= (a^{2})^{2}-2a^{2}b^{2}+(b^{2})^{2}

= (a^{2}-b^{2})^{2}

Now,

= [(a–b) (a+b)]^{2}

= (a – b)^{2} (a + b)^{2}

**5. Factorise the following expressions.**

**(i) p ^{2}+6p+8**

**Solution:**

We can observe that, 8 = 4×2 and 4+2 = 6

Now, we split 6p

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

Taking Common terms,

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

= (p+2) (p+4)

**(ii) q ^{2}–10q+21**

**Solution:**

We can observe that, 21 = -7×-3 and -7+(-3) = -10

Now, split 10q

= q^{2}–3q-7q+21

Taking common terms:

= q(q–3)–7(q–3)

= (q–7) (q–3)

**(iii) p ^{2}+6p–16**

**Solution:**

We can observe that, -16 = -2×8 and 8+(-2) = 6

Now, split 6p

= p^{2}–2p+8p–16

Taking common terms:

= p(p–2) + 8(p–2)

= (p+8) (p–2)

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