ALGEBRAIC IDENTITIES WORKSHEET

1. Expand : (3x + 4y)²

2. Expand : (2p - 3q)²

3. Expand : (5x + 4y)(5x - 4y)

4. Expand : (m + 3)(m + 5)

5. Expand : (m - 7)(m + 2)

6. Expand : (a - b + c)²

7. Expand : (a + b - c)²

8. Expand : (a - b - c)²

9. Expand : (2x + 3y + 4z)²

10. Find the area of square whose side length is (3p + 2q - 4r)

11. Expand (x + 4)(x + 5)(x + 6)

12. Expand (3x - 1)(3x + 2)(3x - 4)

13. Expand (x - y)³

14. Expand (5a - 3b)³

15. Find the product of :

(2x + 3y + 4z)(4x² + 9y² + 16z² - 6xy - 12yz - 8xz)

16. Evaluate : 10³ - 15³ + 5³.

1. Answer :

(3x + 4y)²

We have 

(a + b)² = a² + 2ab + b²

Substitute a = 3x and b = 4y.

(3x + 4y)² = (3x)² + 2(3x)(4y) + (4y)²

= 3²x² + 2(3x)(4y) + 4²y²

= 9x² + 24xy + 16y²

2. Answer :

(2p - 3q)²

We have

(a - b)² = a² - 2ab + b²

Substitute a = 2p and b = 3q.

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

= 2²p² - 12pq + 3²q²

= 4p² - 12pq + 9q²

3. Answer :

(5x + 4y)(5x - 4y)

We have 

(a + b)(a - b) = a² - b²

Substitute a = 5x and b = 4y.

(5x + 4y)(5x - 4y) = (5x)² - (4y)²

= 5²x² - 4²y²

= 25x² - 16y²

4. Answer :

(m + 3)(m + 5)

We have 

(x + a)(x + b) = x² + (a + b)x + ab

Substitute x = m, a = 3 and b = 5.

(m + 3)(m + 5) = m² + (3 + 5)m + (3)(5)

= x² + 8m + 15

5. Answer :

(m - 7)(m + 2)

We have 

(x + a)(x + b) = x² + (a + b)x + ab

Substitute x = m, a = -7 and b = 7.

(m - 7)(m + 2) = m² + (-7 + 2)m + (-7)(2)

= m² - 5m - 14

6. Answer :

(a - b + c)²

We have 

(a + b + c)² = = a² + b² + c² + 2ab + 2bc + 2ac

Replace 'b' by '-b'.

(a + (-b) + c)² = = a² + (-b)² + c² + 2a(-b) + 2(-b)c + 2ac

(a - b + c)² = = a² + b² + c² - 2ab - 2bc + 2ac

7. Answer :

(a + b - c)²

We have 

(a + b + c)² = = a² + b² + c² + 2ab + 2bc + 2ac

Replace 'c' by '-c'.

(a + b + (-c))² = = a² + b² + (-c)² + 2ab + 2b(-c) + 2a(-c)

(a + b + (-c))² = = a² + b² + c² + 2ab - 2bc - 2ac

8. Answer :

(a - b - c)²

We have 

(a + b + c)² = = a² + b² + c² + 2ab + 2bc + 2ac

Replace 'b' by '-b' and 'c' by '-c'.

(a + (-b) + (-c))² = = a² + (-b)² + (-c)² + 2a(-b) + 2(-b)(-c) + 2a(-c)

(a - b - c)² = = a² + b² + c² - 2ab + 2bc - 2ac

9. Answer :

(2x + 3y + 4z)²

We have 

(a + b + c)² = = a² + b² + c² + 2ab + 2bc + 2ac

Substitute a = 2x, b = 3y and c = 4z.

(2x + 3y + 4z)² = (2x)² + (3y)² + (4z)² + 2(2x)(3y) + 2(3y)(4z) + 2(2x)(4z)

= 2²x² + 3²y² + 4²z² + 12xy + 24yz + 16xz

= 4x² + 9y² + 16z² + 12xy + 24yz + 16xz

10. Answer :

Area of Square :

= side x side

= (3p + 2q - 4r)(3p + 2q - 4r)

= (3p + 2q - 4r)²

We know that

(a + b + c)² = = a² + b² + c² + 2ab + 2bc + 2ac

Substitute a = 3p, b = 2q and c = -4r.

(3p + 2q + (-4r)² = (3p)² + (2q)² + (-4r)² + 2(3p)(2q) + 2(2q)(-4r) + 2(3p)(-4r)

= 9p² + 4q² + 16r² + 12pq - 16qr - 24pr

Therefore, Area of Square = [9p² + 4q² + 16r² + 12pq - 16qr - 24pr] square units.

11. Answer :

(x + 4)(x + 5)(x + 6)

We know that

(x + a)(x + b)(x + c) = x³ + (a + b + c)x² + (ab + bc + ac)x + abc

Substitute a = 4, b = 5 and c = 6.

(x + 4)(x + 5)(x + 6) = x³ + (4 + 5 + 6)x² + [(4)(5) + (5)(6) + (4)(6)]x + (4)(5)(6)

= x³ + 15x² + (20 + 30 + 24)x + 120

= x³ + 15x² + 74x + 120

12. Answer :

(3x - 1)(3x + 2)(3x - 4).

We know that

(x + a)(x + b)(x + c) = x³ + (a + b + c)x² + (ab + bc + ac)x + abc

Substitute x = 3x, a = -1, b = 2 and c = -4.

(3x - 1)(3x + 2)(3x - 4) 

= (3x)³ + (-1 + 2 - 4)(3x)² + [(-1)(2) + (2)(-4) + (-1)(-4)](3x) + (-1)(2)(-4)

= 27x³ + (-3)(9x²) + (-2 - 8 + 4)(3x) + 8

= 27x³ - 27x² + (-6)(3x) + 8

= 27x³ - 27x² - 18x + 8

13. Answer :

(x - y)³

We know that

(x + y)³ = x³ + 3x²y + 3xy² + y³

Replace 'y' by '-y'.

(x + (-y))³ = x³ + 3x²(-y) + 3x(-y)² + (-y)³

 = x³ - 3x²y + 3xy² - y³

or

=  = x³ - 3xy(x - y) - y³

14. Answer :

(5a - 3b)³

We know that

(x + y)³ = x³ + 3x²y + 3xy² + y³

We know that

(x + y)³ = x³ + 3x²y + 3xy² + y³

Substitute x = 5a, and y = -3b.

(5a - 3b)³ = (5a)³ + 3(5a)²(-3b) + 3(5a)(-3b)² + (-3b)³

= 5³a³ + 3(25a²)(-3b) + 3(5a)(9b²) + (-27b³)

= 125a³ - 225a²b + 135ab² - 27b³

15. Answer :

(2x + 3y + 4z)(4x² + 9y² + 16z² - 6xy - 12yz - 8xz)

We know that

(a + b + c)(a² + b² + c² - ab - bc - ac) = a³ + b³ + c³ - 3abc

Substitute a = 2x, b = 3y and c = 4z.

(2x + 3y + 4z)(4x² + 9y² + 16z²  - 6xy - 12yz - 8xz)

= (2x)³ + (3y)³ + (4z)³ - 3(2x)(3y)(4z)

= 8x³ + 27y³ + 64z³ - 72xyz

16. Answer :

10³ - 15³ + 5³

We know that, If (a + b + c) = 0, then a³ + b³ + c³ = 3abc.

If a = 10, b = -15 and c = 5, then

a + b + c = 10 - 15 + 5 = 0

Therefore,

10³ + (-15)³ + 5³ = 3(10)(-15)(5)

10³ - 15³ + 5³ = -2250