OPERATIONS WITH POLYNOMIALS

Simplify the following polynomials.

Problem 1 :

20m⁵ + 4m⁵

Solution :

20m⁵ + 4m⁵

= (20 + 4)m⁵

= 24m⁵

Problem 2 :

(5p² – 3) + (2p² – 3p³)

Solution :

(5p² – 3) + (2p² – 3p³)

= 5p² – 3 + 2p² – 3p³

= -3p³ + 7p² - 3

Problem 3 :

(-9v² – 8u) + (-2uv – 2u² + v²) + (-v² + 4uv)

Solution :

(-9v² – 8u) + (-2uv – 2u² + v²) + (-v² + 4uv)

= -9v² – 8u - 2uv – 2u² + v² - v² + 4uv

= -2u² – 8u + 2uv – 9v²

Problem 4 :

20m⁵ - 4m⁵

Solution :

20m⁵ - 4m⁵

= (20 – 4)m⁵

= 16m⁵

Problem 5 :

(8n – 3n⁴ + 10n²) – (3n² + 11n⁴ – 7)

Solution :

(8n – 3n⁴ + 10n²) – (3n² + 11n⁴ – 7)

= 8n – 3n⁴ + 10n² – 3n² - 11n⁴ + 7

= -14n⁴ + 7n² + 8n + 7

Problem 6 :      

(4x² + 7x³y²) – (-6x² – 7x³y² – 4x) – (10x + 9x²)

Solution :

(4x² + 7x³y²) – (-6x² – 7x³y² – 4x) – (10x + 9x²)

= 4x² + 7x³y² + 6x² + 7x³y² + 4x – 10x - 9x²

= 4x² + 6x² – 9x² + 7x³y² + 7x³y² + 4x – 10x

= x² + 14x³y² – 6x

Multiplying Polynomials

Problem 1 :

5x(7x – 3)

Solution :

5x(7x – 3)

= 5x ⋅ 7x – 5x ⋅ 3

= 35x² – 15x

Problem 2 :

7a(3a + b)

Solution :

7a(3a + b)

= 7a ⋅ 3a + 7a ⋅ b

= 21a² + 7ab

Problem 3 :

5a(6a⁵ + 8a³ – 4)

Solution :

5a(6a⁵ + 8a³ – 4)

= 5a ⋅ 6a⁵ + 5a ⋅ 8a³ – 5a ⋅ 4

= 30a⁶ + 40a⁴ – 20a

Problem 4 :

2x⁵(4x² – 7x + 2)

Solution :

2x⁵(4x² – 7x + 2)

= 2x⁵ ⋅ 4x² – 2x⁵ ⋅ 7x + 2x⁵ ⋅ 2

= 8x⁷ – 14x⁶ + 4x⁵

Problem 5 :

a²b³c⁸(a⁴b⁸ – b³c⁹)

Solution :

a²b³c⁸(a⁴b⁸ – b³c⁹)

= a²b³c⁸ ⋅ a⁴b⁸ - a²b³c⁸ ⋅ b³c⁹

= a⁶b¹¹c⁸ – a²b⁶c¹⁷  

Problem 6 :

(1/2)⋅(4x² – 8x + 10)

Solution :

= (1/2)⋅(4x² – 8x + 10)

= (1/2) ⋅ 2(2x² – 4x + 5)

= (2x² – 4x + 5)

Simplify :

Problem 1 :

(4n + 8)/2

Solution :

(4n + 8)/2

= 4n/2 + 8/2

= 2n + 4

= 2(n + 2)

Problem 2 :

(-9g + 12)/-3

Solution :

(-9g + 12)/-3

= (-9g/-3) + (12/-3)

= 3g – 4`

Problem 3 :

(16x² – 12x + 8)/4

Solution :

(16x² – 12x + 8)/4

= 16x²/4 – 12x/4 + 8/4

= 4x² – 3x + 2

Problem 4 :

(27w² + 18w - 36)/-9

Solution :

(27w² + 18w - 36)/-9

= (27w²/-9) + (18w/-9) – (36/-9)

= -3w² – 2w + 4

Problem 5 :

(3x² + 6x)/2x

Solution :

(3x² + 6x)/2x

= 3x²/2x + 6x/2x

= (3/2)x + 3

Problem 6 :

(25s³ + 12s)/5s

Solution :

(25s³ + 12s)/5s

= 25s³/5s + 12s/5s

= 5s² + 12s/5