**Exercise 4.1**

**Question 1: Evaluate each of the following using identities:**

**(i) (2x – 1/x)**

^{2}**(ii) (2x+y) (2x – y)**

**(iii) (a**

^{2}b-b^{2}a)^{2}**(iv) (a – 0.1) (a + 0.1)**

**(v) (1.5.x**

^{2}– 0.3y^{2}) (1.5x^{2}+ 0.3y^{2})**Solution: 1**

^{2}

^{2}= a

^{2}+ b

^{2}– 2ab

^{2}

^{2}+ (1/x)

^{2}– 2 (2x) 1/x

^{2}+ 1/x

^{2}– 4

^{2}= a

^{2}+ b

^{2}– 2ab

^{2}–(y)

^{2}

^{2}– y

^{2}

^{2}b – b

^{2}a)

^{2}

^{2}= a

^{2}+ b

^{2}– 2ab

^{2}b –b

^{2}a)

^{2}

^{2}b)

^{2}+(b

^{2}a)

^{2}– 2 (a

^{2}b)(b

^{2}a)

^{4}b

^{2}+b

^{4}a

^{2}–2a

^{3}b

^{3}

^{2}= a

^{2}+ b

^{2}– 2ab

^{2}– (0.1)

^{2}

^{2}–0.01

^{2}– 0.3y

^{2}) (1.5x

^{2}+ 0.3y

^{2})

^{2}= a

^{2}+ b

^{2}– 2ab

^{2}– 0.3y

^{2}) (1.5x

^{2}+ 0.3y

^{2})

^{2})

^{2}– (0.3y

^{2})

^{2}

^{4}– 0.09y

^{4}

**Question 2: Evaluate each of the following using identities:**

**(i) (399)**

^{2}**(ii) (0.98)**

^{2}**(iii) 991 x 1009**

**(iv) 117 x 83**

**Solution: 2**

^{2}=(400-1)

^{2}

^{2}-(1)

^{2}-2×400×1

^{2}=a

^{2}+b

^{2}-2ab

^{2}+(1)

^{2}-2×400 ×1

^{2}is 159201

^{2}=(1-0.02)

^{2}

^{2}+(0.02)

^{2}-2×1×0.02

^{2}=a

^{2}+b

^{2}-2ab

^{2}is 0.9604

^{2}=a

^{2}+b

^{2}-2ab

^{2}-(9)

^{2}

^{2}=a

^{2}+b

^{2}-2ab

^{2}-(17)

^{2}

**Question 3: Simplify each of the following:**

**(i) 175 x 175 +2 x 175 x 25 + 25 x 25**

**(ii) 322 x 322 – 2 x 322 x 22 + 22 x 22**

**(iii) 0.76 x 0.76 + 2 x 0.76 x 0.24 + 0.24 x 0.24**

**Solution: 3**

^{2}+2 (175) (25)+(25)

^{2}

^{2}+b

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

^{2}

^{2}

^{2}

^{2}– 2 x 322 x 22 + (22)

^{2}

^{2}+b

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

^{2}

^{2}

^{2}

^{2}+b

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

^{2}

^{2}

^{2}

**Question 4: If x+1/x=11, find the value of x**

^{2}+1/x^{2}.**Solution: 4**

**Question 5: If x – 1/x=-1, find the value of x**

^{2}+1/x^{2}**Solution: 5**

**Exercise 4.2**

**Question 1: Write the following in the expanded form:**

**(i) (a+2b+c)**

^{2}**(ii) (2a-3b-c)**

^{2}**(iii) (-3x+y+z)**

^{2}**(iv) (m+2n-5p)**

^{2}**(v) (2+x-2y)**

^{2}**(vi) (a**

^{2}+b^{2}+c^{2})^{2}**(vii) (ab+bc+ca)**

^{2}**(viii) (x/y+y/z+z/x )**

^{2}**(ix) (a/bc+ b/ac+ c/ab )**

^{2}**(x) (x+2y+4z)**

^{2}**(xi) (2x-y+z)**

^{2}**(xii) (-2x+3y+2z)**

^{2}**Solution: 1**

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+(2b)

^{2}+c

^{2}+2a×2b+2ac+2×2b×c

^{2}+ 4b

^{2}+ c

^{2}+4ab+2ac+4bc

^{2}is a

^{2}+ 4b

^{2}+ c

^{2}+4ab+2ac+4bc.

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}

^{2}+(-3b)

^{2}+(-c)

^{2}+2(2a)(-3b)+2(-3b)(-c)+2(2a)(-c)

^{2}+9b

^{2}+c

^{2}-12ab + 6bc - 4ca

^{2}is 4a

^{2}+9b

^{2}+c

^{2}-12ab + 6bc - 4ca.

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+y

^{2}+ z

^{2}+2(-3x)y + 2yz + 2(-3x)z

^{2}+y

^{2}+z

^{2}- 6xy + 2yz -6xz

^{2}is 9x

^{2}+y

^{2}+z

^{2}- 6xy + 2yz -6xz

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+(2n)

^{2 }+(-5p)

^{2}+ 2m × 2n + (2×2n×-5p) + 2m × -5p

^{2}+ 4n

^{2}+ 25p

^{2}+ 4mn - 20np - 10pm

^{2}is m

^{2}+ 4n

^{2}+ 25p

^{2}+ 4mn - 20np - 10pm.

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+ x

^{2}+ (-2y)

^{2}+ 2(2)(x) + 2(x)(-2y) + 2(2)(-2y)

^{2}+ 4y

^{2}+ 4 x - 4xy - 8y

^{2}is 4+x

^{2}+4y

^{2}+4x-4xy-8y.

^{2}+b

^{2}+c

^{2})

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2})

^{2}+(b

^{2})

^{2}+(c

^{2})

^{2}+2a

^{2}b

^{2}+2b

^{2}c

^{2}+2a

^{2}c

^{2}

^{4}+b

^{4}+c

^{4}+2a

^{2}b

^{2}+2b

^{2}c

^{2}+ 2c a

^{2}

^{2}+b

^{2}+c

^{2})

^{2}is a

^{4}+b

^{4}+c

^{4}+2a

^{2}b

^{2}+2b

^{2}c

^{2}+2c

^{2}a

^{2}.

**Question 2: Simplify**

**(i) (a + b + c)**

^{2}+ (a − b + c)^{2}**(ii) (a + b + c)**

^{2}− (a − b + c)^{2}**(iii) (a + b + c)**

^{2}+ (a – b + c)^{2}+ (a + b − c)^{2}**(iv) (2x + p − c)**

^{2}− (2x − p + c)^{2}**(v) (x**

^{2}+ y^{2}− z^{2})^{2}− (x^{2}− y^{2}+ z^{2})^{2}**Solution: 2**

^{2}+ (a − b + c)

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+ b

^{2}+ c

^{2}+ 2ab+2bc+2ca) + (a

^{2}+ (−b)

^{2}+ c

^{2}−2ab−2bc+2ca)

^{2}+ 2b

^{2}+ 2c

^{2}+ 4ca

^{2}+ (a − b + c)

^{2}is 2a

^{2}+ 2b

^{2}+ 2c

^{2}+ 4ca.

^{2}− (a − b + c)

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+ b

^{2}+ c

^{2}+ 2ab + 2bc + 2ca) − (a

^{2}+ (−b)

^{2}+ c

^{2}−2ab − 2bc + 2ca)

^{2}+ b

^{2}+ c

^{2}+ 2ab + 2bc + 2ca − a

^{2}− b

^{2}− c

^{2}+ 2ab + 2bc − 2ca

^{2}− (a − b + c)

^{2}is 4ab + 4bc.

^{2}+ (a – b + c)2 + (a + b − c)

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+ b

^{2}+ c

^{2}+ 2ab + 2bc + 2ca + (a

^{2}+ b

^{2}+ c)

^{2}− 2ab − 2cb + 2ca) + (a

^{2}+ b

^{2}+ c

^{2}+ 2ab − 2bc – 2ca)

^{2}+ 3b

^{2}+ 3c

^{2}+ 2ab − 2bc + 2ca

^{2}+ (a – b + c)

^{2}+ (a + b − c)

^{2}is 3a

^{2}+ 3b

^{2}+ 3c

^{2}+ 2ab − 2bc + 2ca.

^{2}− (2x − p + c)

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+p

^{2}+(-c)

^{2}+2×2x×p+2×p×-c+2×2x×-c]-[(-2x)

^{2}+p

^{2}+c

^{2}+2×-2x×p+2×p×c+2×-2x×c]

^{2}+p

^{2}+c

^{2}+4xp-2pc-4cx] − [4x

^{2}+p

^{2}+ c

^{2}- 4xp-2pc+4cx]

^{2}+p

^{2}+c

^{2}+4xp-2pc-4cx-4x

^{2}-p

^{2}- c

^{2}+ 4xp + 2pc- 4cx

^{2}− (2x − p + c)

^{2}is 8(xp - xc).

^{2}+ y

^{2}− z

^{2})

^{2}− (x

^{2}− y

^{2}+ z

^{2})

^{2}

^{2}=x

^{2}+y

^{2}+z

^{2}+2xy+2yz+2xz

^{2}+ y

^{2}+ (−z)

^{2})

^{2}− (x

^{2}− y

^{2}+ z

^{2})

^{2}

^{4}+ y

^{4}+ z

^{4}+ 2x

^{2}y

^{2}– 2y

^{2}z

^{2}– 2x

^{2}z

^{2}− [x

^{4}+ y

^{4}+ z

^{4}− 2x

^{2}y

^{2}− 2y

^{2}z

^{2}+ 2x

^{2}z

^{2}]

^{2}y

^{2}– 4z

^{2}x

^{2}

^{2}+ y

^{2}− z

^{2})

^{2}− (x

^{2}− y

^{2}+ z

^{2})2 is 4x

^{2}y

^{2}– 4z

^{2}x

^{2}.

**Question 3: If a + b + c = 0 and a**

^{2}+ b^{2}+ c^{2}= 16, find the value of ab + bc + ca.**Solution: 3**

^{2}+ b

^{2}+ c

^{2}= 16

^{2}= (0)

^{2}

^{2}= a

^{2}+ b

^{2}+ c

^{2}+ 2(ab + bc + ca)

^{2}+ b

^{2}+ c

^{2}+ 2(ab + bc + ca) = 0

**Exercise 4.3**

**Question 1: Find the cube of each of the following binomial expressions:**

**(i) (1/x + y/3)**

**(ii) (3/x – 2/x**

^{2})**(iii) (2x + 3/x)**

**(iv) (4 – 1/3x)**

**Solution:**

**Question 2: Simplify each of the following:**

**Solution:**

**Question 3: If a + b = 10 and ab = 21, find the value of a**

^{3}+ b^{3}.**Solution:**

^{3}= (10)

^{3}

^{3}= a

^{3}+ b

^{3}+ 3ab (a + b)

^{3}+ b

^{3}is 370.

**Question 4: If a – b = 4 and ab = 21, find the value of a**

^{3}– b^{3}.**Solution:**

^{3}= (4)

^{3}

^{3}= a

^{3}+ b

^{3}+ 3ab (a + b)

^{3}– b

^{3}– 3ab (a – b) = 64

^{3}– b

^{3}– 3 × 21 x 4 = 64

^{3}– b

^{3}– 252 = 64

^{3}– b

^{3}= 64 + 252

^{3}– b

^{3}= 316

^{3}– b

^{3}is 316

**Question 5: If x + 1/x= 5, find the value of x**

^{3}+ 1/x^{3}.**Solution:**

**Question 6: If x – 1/x= 7, find the value of x**

^{3}– 1/x^{3}.**Solution:**

**Question 7: If x – 1/x = 5, find the value of x**

^{3}– 1/x^{3}.**Solution:**

**Question 8: If (x**

^{2}+ 1/x^{2}) = 51, find the value of .**Solution:**

**Question 9: If (x**

^{2}+1/x^{2})= 98, find the value of (x^{3}+1/x^{3})**Solution:**

**Question 10: If 2x + 3y = 13 and xy = 6, find the value of 8x**

^{3}+ 27y^{3}.**Solution:**

^{3}= (13)

^{3}

^{3}= a

^{3}+ b

^{3}+ 3ab (a + b)

^{3}+ (3y)

^{3}+ 3 × (2x)(3y) (2x + 3y) = 13 × 13 × 13

^{3}+ 27y

^{3}+ 18xy (2x + 3y) = 2197

^{3}+ 27y

^{3}+ 18 x 6 x 13 = 2197

^{3}+ 27y

^{3}+ 1404 = 2197

^{3}+ 27y

^{3}= 2197 – 1404

^{3}+ 27y

^{3}= 793

^{3}+ 27y

^{3}is 793.

**Question 11: If 3x – 2y= 11 and xy = 12, find the value of 27x**

^{3}– 8y^{3}.**Solution:**

^{3}= (11)

^{3}

^{3}= a

^{3}- b

^{3}- 3ab (a - b)

^{3}– (2y)

^{3}– 3 (3x)( 2y) (3x – 2y) = 1331

^{3}– 8y

^{3}– 18xy (3x -2y) = 1331

^{3}– 8y

^{3}– 18 x 12 x 11 = 1331

^{3}– 8y

^{3}– 2376 = 1331

^{3}– 8y

^{3}= 1331 + 2376

^{3}– 8y

^{3}= 3707

^{3}– 8y

^{3}is 3707.

**Exercise 4.4**

**Question 1: Find the following products:**

**(i) (3x + 2y)(9x**

^{2}– 6xy + 4y^{2})**(ii) (4x – 5y)(16x**

^{2}+ 20xy + 25y^{2})**(iii) (7p**

^{4}+ q)(49p^{8}– 7p4q + q^{2})**(iv) (x/2+ 2y) (x**

^{2}/4– xy + 4y^{2})**(v) (3/x – 5/y)(9/x**

^{2}+ 25/y^{2}+ 15/xy)**(vi) (3 + 5/x)(9 – 15/x + 25/x**

^{2})**(vii) (2/x + 3x)(4/x**

^{2}+ 9x^{2}– 6)**(viii) (3/x – 2x**

^{2})(9/x^{2}+ 4x^{4}– 6x)**(ix) (1 – x)(1 + x + x**

^{2})**(x) (1 + x)(1 – x + x**

^{2})**(xi) (x**

^{2}– 1)(x^{4}+ x^{2}+1)**(xii) (x**

^{3}+ 1)(x^{6}– x^{3}+ 1)**Solution:**

^{2}– 6xy + 4y

^{2})

^{2}– (3x)(2y) + (2y)

^{2}]

^{3}+ b

^{3}= (a + b)(a

^{2}+ b

^{2}– ab)

^{3}+ (2y)

^{3}

^{3}+ 8y

^{3}

^{2}– 6xy + 4y

^{2}) is 27x

^{3}+ 8y

^{3}.

^{2}+ 20xy + 25y

^{2})

^{2}+ (4x)(5y) + (5y)

^{2})]

^{3}– b

^{3}= (a – b)(a

^{2}+ b

^{2}+ ab)

^{3}– (5y)

^{3}

^{3}– 125y

^{3}

^{2}+ 20xy + 25y

^{2}) is 64x

^{3}– 125y

^{3}.

^{4}+ q)(49p

^{8}– 7p

^{4}q + q

^{2})

^{4}+ q)[(7p

^{4})

^{2}– (7p

^{4})(q) + (q)

^{2})]

^{3}+ b

^{3}= (a + b)(a

^{2}+ b

^{2}– ab)

^{2}and b=q

^{4})

^{3}+ (q)

^{3}

^{12}+ q

^{3}

^{4}+ q)(49p

^{8}– 7p

^{4}q + q

^{2}) is 343 p

^{12}+ q

^{3}.

^{2})

^{3}– b

^{3}= (a – b)(a

^{2}+ b

^{2}+ ab)

^{2})

^{2})]

^{3}– (x)

^{3}

^{3}

^{2}) is 1 – x

^{3}.

^{2})

^{3}+ b

^{3}= (a + b)(a

^{2}+ b

^{2}– ab)

^{2})

^{2})]

^{3}+ (x)

^{3}

^{3}

^{2}) is 1 + x

^{3}

^{2}– 1)(x

^{4}+ x

^{2}+1)

^{3}– b

^{3}= (a – b)(a

^{2}+ b

^{2}+ ab)

^{2}– 1)[(x

^{2})

^{2}– (1)

^{2}+ (x

^{2})(1)]

^{2}– 1)[(x

^{2})

^{2}– (1)

^{2}+ x

^{2}]

^{2})

^{3}– (1)

^{3}

^{6}– 1

^{2}– 1)(x

^{4}+ x

^{2}+1) is x

^{6}– 1.

^{3}+ 1)(x

^{6}– x

^{3}+ 1)

^{3}+ b

^{3}= (a + b)(a

^{2}+ b

^{2}– ab)

^{3}+ 1)[(x

^{3})

^{2}– (x

^{3})(1) + 12]

^{3}+ 1)[(x

^{3})

^{2}– x

^{3}+ 12]

^{3})

^{3}+ 1

^{3}

^{9}+ 1

^{3}+ 1)(x

^{6}– x

^{3}+ 1) is x

^{9}+ 1.

**Question 2: If x = 3 and y = -1, find the values of each of the following using in identity:**

**(i) (9y**

^{2}– 4x^{2})(81y^{4}+ 36x^{2}y^{2}+ 16x^{4})**(ii) (3/x – x/3)(x**

^{2}/9 + 9/x^{2}+ 1)**(iii) (x/7 + y/3)(x**

^{2}/49 + y^{2}/9 – xy/21)**(iv) (x/4 – y/3)(x**

^{2}/16 + xy/12 + y^{2}/9)**(v) (5/x + 5x)(25/x**

^{2}– 25 + 25x^{2})**Solution:**

^{2}– 4x

^{2})(81y

^{4}+ 36x

^{2}y

^{2}+ 16x

^{4})

^{2}– 4x

^{2}) [(9y

^{2})

^{2}+ 9y

^{2}x 4x

^{2}+ (4x

^{2})

^{2}

^{3}+ b

^{3}= (a + b)(a

^{2}+ b

^{2}– ab)

^{2})

^{3}– (4x

^{2})

^{3}

^{6}– 64x

^{6}

^{6}– 64(3)

^{6}

^{2}– 4x

^{2})(81y

^{4}+ 36x

^{2}y

^{2}+ 16x

^{4}) is - 45927.

**Question 3: If a + b = 10 and ab = 16, find the value of a**

^{2}– ab + b^{2}and a^{2}+ ab + b^{2}.**Solution:**

^{2}= (10)

^{2}

^{2}= a

^{2}+ b

^{2}+ 2ab

^{2}+ b

^{2}+ 2ab = 100

^{2}+ b

^{2}+ 2 x 16 = 100

^{2}+ b

^{2}+ 32 = 100

^{2}+ b

^{2}= 100 – 32 = 68

^{2}+ b

^{2}= 68

^{2}– ab + b

^{2}is

^{2}+ b

^{2}– ab

^{2}+ ab + b

^{2}

^{2}+ b

^{2}+ ab

^{2}– ab + b

^{2}is 52 and a

^{2}+ ab + b

^{2}is 84.

**Question 4: If a + b = 8 and ab = 6, find the value of a**

^{3}+ b^{3}.**Solution:**

^{3}= (8)

^{3}

^{3}= a

^{3}+ b

^{3}+ 3ab (a + b)

^{3}+ b

^{3}+ 3ab (a + b) = 512

^{3}+ b

^{3}+ 3 x 6 x 8 = 512

^{3}+ b

^{3}+ 144 = 512

^{3}+ b

^{3}= 512 – 144 = 368

^{3}+ b

^{3}= 368

^{3}+ b

^{3}is 368.

**Exercise 4.5**

**Question 1: Find the following products:**

**(i) (3x + 2y + 2z) (9x**

^{2}+ 4y^{2}+ 4z^{2}– 6xy – 4yz – 6zx)**(ii) (4x – 3y + 2z) (16x**

^{2}+ 9y^{2}+ 4z^{2}+ 12xy + 6yz – 8zx)**(iii) (2a – 3b – 2c) (4a**

^{2}+ 9b^{2}+ 4c^{2}+ 6ab – 6bc + 4ca)**(iv) (3x -4y + 5z) (9x**

^{2}+ 16y^{2}+ 25z^{2}+ 12xy- 15zx + 20yz)**Solution:**

^{2}+ 4y

^{2}+ 4z

^{2}– 6xy – 4yz – 6zx)

^{2}+ (2y)

^{2}+ (2z)

^{2}– 3x x 2y – 2y x 2z – 2z x 3x]

^{3}+ b

^{3}+ c

^{3}- 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca)

^{3}+ (2y)

^{3}+ (2z)

^{3}– 3 x 3x x 2y x 2z

^{3}+ 8y

^{3}+ 8z

^{3}– 36xyz

^{2}+ 4y

^{2}+ 4z

^{2}–6xy–4yz –6zx) is 27x

^{3}+ 8y

^{3}+ 8z

^{3}– 36xyz

^{2}+ 9y

^{2}+ 4z

^{2}+ 12xy + 6yz – 8zx)

^{2}+ (-3y)

^{2}+ (2z)

^{2}– 4x x (-3y) – (-3y) x (2z) – (2z x 4x)]

^{3}+ b

^{3}+ c

^{3}- 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca)

^{3}+ (-3y)

^{3}+ (2z)

^{3}– 3 x 4x x (-3y) x (2z)

^{3}– 27y

^{3}+ 8z

^{3}+ 72xyz

^{2}+ 9y

^{2}+ 4z

^{2}+ 12xy + 6yz – 8zx) is 64x

^{3}– 27y

^{3}+ 8z

^{3}+ 72xyz.

^{2}+ 9b

^{2}+ 4c

^{2}+ 6ab – 6bc + 4ca)

^{2}+ (-3b)

^{2}+ (-2c)

^{2}– 2a x (-3b) – (-3b) x (-2c) – (-2c) x 2a]

^{3}+ b

^{3}+ c

^{3}- 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca)

^{3}+ (-3b)

^{3}+ (-2c)

^{3}-3x 2a x (-3 b) (-2c)

^{3}– 21b

^{3}– 8c

^{3}– 36abc

^{2}+ 9b

^{2}+ 4c

^{2}+ 6ab – 6bc + 4ca) is 8a

^{3}– 21b

^{3}– 8c

^{3}– 36abc.

^{2}+ 16y

^{2}+ 25z

^{2}+ 12xy- 15zx + 20yz)

^{2}+ (-4y)

^{2}+ (5z)

^{2}– 3x x (-4y) -(-4y) (5z) – 5z x 3x]

^{3}+ b

^{3}+ c

^{3}- 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca)

^{3}+ (-4y)

^{3}+ (5z)

^{3}– 3 x 3x x (-4y) (5z)

^{3}– 64y

^{3}+ 125z

^{3}+ 180xyz

^{2}+ 16y

^{2}+ 25z

^{2}+ 12xy – 15zx + 20yz) is 27x

^{3}– 64y

^{3}+ 125z

^{3}+ 180xyz.

**Question 2: If x + y + z = 8 and xy + yz+ zx = 20, find the value of x**

^{3}+ y^{3}+ z^{3}– 3xyz.**Solution:**

^{2 }= a

^{2}+b

^{2}+c

^{2}+2(ab+bc+ca)

^{2}= (8)

^{2}

^{2}+ y

^{2}+ z

^{2}+ 2(xy + yz + zx) = 64

^{2}+ y

^{2}+ z

^{2}+ 2 x 20 = 64

^{2}+ y

^{2}+ z

^{2}+ 40 = 64

^{2}+ y

^{2}+ z

^{2}= 24

^{3}+ b

^{3}+ c

^{3}- 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca)

^{3}+ y

^{3}+ z

^{3}– 3xyz

^{2}+ y

^{2}+ z

^{2}– (xy + yz + zx)]

^{3}+ y

^{3}+ z

^{3}– 3xyz is 32.

**Question 3: If a +b + c = 9 and ab + bc + ca = 26, find the value of a**

^{3}+ b^{3}+ c^{3}– 3abc.**Solution:**

^{2 }= a

^{2}+b

^{2}+c

^{2}+2(ab+bc+ca)

^{2}= (9)

^{2}

^{2}+ b

^{2}+ c

^{2}+ 2 (ab + bc + ca) = 81

^{2}+ b

^{2}+ c

^{2}+ 2 x 26 = 81

^{2}+ b

^{2}+ c

^{2}+ 52 = 81

^{2}+ b

^{2}+ c

^{2}= 29

^{3}+ b

^{3}+ c

^{3}- 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca)a

^{3}+ b

^{3}+ c

^{3}- 3abc

^{2}+ b

^{2}+ c

^{2}- ab - bc - ca) ………………..(iii)

^{3}+ b

^{3}+ c

^{3}– 3abc is 27

**Exercise VSAQs..........................**

**Question 1: If x + 1/x = 3, then find the value of x**

^{2}+ 1/x^{2}.**Solution:**

^{2}= (3)

^{2}

^{2}= a

^{2}+ 2ab + b

^{2}

^{2}+1/x

^{2}+ 2x×1/x=9

^{2}+1/x

^{2}+ 2=9

^{2}+1/x

^{2}=9-2

^{2}+1/x

^{2}=7

^{2}+1/x

^{2}is 7.

**Question 2: If x + 1/x = 3, then find the value of x**

^{6}+ 1/x^{6}.**Solution:**

^{2}= (3)

^{2}

^{2}= a

^{2}+ 2ab + b

^{2 }

^{2}+1/x

^{2}+ 2x×1/x=9

^{2}+1/x

^{2}+ 2=9

^{2}+1/x

^{2}=9-2

^{2}+1/x

^{2}=7 ………………(i)

^{2}+1/x

^{2})

^{3}= (7)

^{3}

^{3}+ b

^{3}+ 3(a + b)

^{6}+ 1/x

^{6}+ 3 (x

^{2}+ 1/x

^{2}) = 343

^{6}+ 1/x

^{6}+3 x 7=343

^{6}+ 1/x

^{6}= 322

^{6}+ 1/x

^{6}is 322.

**Question 3: If a + b = 7 and ab = 12, find the value of a**

^{2}+ b^{2}.**Solution:**

^{2}= (7)

^{2}

^{2}= a

^{2}+ 2ab + b

^{2 }

^{2}+ b

^{2}+ 2ab = 49

^{2}+ b

^{2}+ 2 x 12 = 49

^{2}+ b

^{2}+ 24 = 49

^{2}+ b

^{2}= 25

^{2}+ b

^{2}is 25.

**Question 4: If a – b = 5 and ab = 12, find the value of a**

^{2}+ b^{2}.**Solution:**

^{2}= (5)

^{2}

^{2}= a

^{2}- 2ab + b

^{2}

^{2}+ b

^{2}– 2ab = 25

^{2}+ b

^{2}– 2 x 12 = 25

^{2}+ b

^{2}– 24 = 25

^{2}+ b

^{2}= 49

^{2}+ b

^{2}is 49.