PROBLEMS ON ADDING AND SUBTRACTING POLYNOMIALS

If P(x) = x³ + 2x² – 4x + 1, Q(x) = x² + 3x – 2 and R(x) = x³ + 3x – 1, determine each of the following.

Problem 1 :

P(x) + Q(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

Q(x) = x² + 3x – 2

P(x) + Q(x) = [x³ + 2x² – 4x + 1] + [x² + 3x – 2]

= x³ + 2x² – 4x + 1 + x² + 3x – 2

By combining like terms,

= x³ + 2x² + x² – 4x + 3x + 1 – 2

= x³ + 3x² – x - 1

Problem 2 :

P(x) + R(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

R(x) = x³ + 3x – 1

P(x) + R(x) = [x³ + 2x² – 4x + 1] + [x³ + 3x – 1]

= x³ + 2x² – 4x + 1 + x³ + 3x – 1

By combining like terms,

= x³ + x³ + 2x² – 4x + 3x

= 2x³ + 2x² – x

Problem 3 :

P(x) - Q(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

Q(x) = x² + 3x – 2

P(x) - Q(x) = [x³ + 2x² – 4x + 1] - [x² + 3x – 2]

= x³ + 2x² – 4x + 1 - x² - 3x + 2

By combining like terms,

= x³ + 2x² - x² – 4x - 3x + 1 + 2

= x³ + x² – 7x + 3

Problem 4 :

P(x) - R(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

R(x) = x³ + 3x – 1

P(x) - R(x) = [x³ + 2x² – 4x + 1] - [x³ + 3x – 1]

= x³ + 2x² – 4x + 1 - x³ - 3x + 1

By combining like terms,

= 2x² – 4x - 3x + 1 + 1

= 2x² - 7x + 2

Problem 5 :

R(x) - P(x)

Solution :

Given, R(x) = x³ + 3x – 1

P(x) = x³ + 2x² – 4x + 1

R(x) - P(x) = [x³ + 3x – 1] – [x³ + 2x² – 4x + 1]

= x³ + 3x – 1 - x³ - 2x² + 4x – 1

By combining like terms,

= 3x + 4x - 2x² - 1 - 1

= -2x² + 7x - 2

Problem 6 :

P(x) + Q(x) + R(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

Q(x) = x² + 3x – 2

R(x) = x³ + 3x – 1

P(x) + Q(x) + R(x) = [x³ + 2x² – 4x + 1] + [x² + 3x – 2] + [x³ + 3x – 1]

= x³ + 2x² – 4x + 1 + x² + 3x – 2 + x³ + 3x - 1

By combining like terms,

= x³ + x³ + 2x² + x² – 4x + 3x + 3x + 1 – 2 - 1

= 2x³ + 3x² + 2x - 2

Problem 7 :

P(x) - Q(x) + R(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

Q(x) = x² + 3x – 2

R(x) = x³ + 3x – 1

P(x) - Q(x) + R(x) = [x³ + 2x² – 4x + 1] - [x² + 3x – 2] + [x³ + 3x – 1]

= x³ + 2x² – 4x + 1 - x² - 3x + 2 + x³ + 3x - 1

By combining like terms,

= x³  + x³ + 2x² - x²  - 4x - 3x + 3x + 1 + 2 - 1

= 2x³ + x²– 4x + 2

Problem 8 :

P(x) - Q(x) – R(x)

Solution :

Given, P(x) = x³ + 2x² – 4x + 1

Q(x) = x² + 3x – 2

R(x) = x³ + 3x – 1

P(x) - Q(x) - R(x) = [x³ + 2x² – 4x + 1] - [x² + 3x – 2] - [x³ + 3x – 1]

= x³ + 2x² – 4x + 1 - x² - 3x + 2 - x³ - 3x + 1

By combining like terms,

= x³ - x³ + 2x² - x² – 4x - 3x - 3x + 1 + 2 + 1

= x² - 10x + 4