operations with polynomials

A. ADDITION

To add polynomials, we only have to add their monomials.

Example:

(2x³ + 3x² – 5x + 2) + (5x² – 3x + 21) = 2x³ + 3x² – 5x +2 + 5x² - 3x + 21 = 2x³ + 8x² – 8x + 23

B. SUBTRACTION

The opposite of a polynomial is another polynomial with the opposite monomials.

For example:
Op (3x⁴ – 2x² + x – 1) = -3x⁴ +2x² – x + 1

To subtract polynomials, we only have to add the first polynomial plus the opposite of the second polynomial.

Example:
(2x³ + 3x² – 5x + 2) - (5x² – 3x + 21) = 2x³ + 3x² – 5x +2 - 5x² + 3x - 21 = 2x³ - 2x² – 2x - 19

C. MULTIPLICATION

    (5x³ – 3x² + 1) · 2x² = (5x³ · 2x²) – (3x² · 2x²) + +(1 · 2x²) =  10x⁵ – 6x⁴ + 2x²

(5x³ – 3x² + 1) · (2x² + 3) =  (5x³ · 2x²) – (3x² · 2x²) + (1 · 2x²) + (5x³ · 3) – (3x² · 3) + (1 · 3) = 10x⁵ – 6x⁴ + 2x² + 15x³ – 9x² + 3 = 10x⁵ – 6x⁴ +15x³ – 7x² +3

Exercise. If A = x³ - 3x² +5x - 5, B = x³ -3x² - 5x +5, C = x² + 2. Calculate:

a) A + B

b) A - B

c) A · C

Solutions: a) 2x³ - 6x²; b) 10x -10; c) x⁵ - 3x⁴ + 7x³ - 11x² + 10x -10

C. MULTIPLICATION

To multiply a polynomial by a monomial, we have to multiply the monomial by each monomial in the polynomial (distributive property).

Example:
(5x³ – 3x² + 1) · 2x² = (5x³ · 2x²) – (3x² · 2x²) + +(1 · 2x²) = 10x⁵ – 6x⁴ + 2x²

To multiply two polynomials, we have to multiply each monomial in one polynomial by the other polynomial.

Example:

(5x³ – 3x² + 1) · (2x² + 3) = (5x³ · 2x²) – (3x² · 2x²) + (1 · 2x²) + (5x³ · 3) -(3x² · 3) + (1 · 3) = 10x⁵ – 6x⁴ + 2x² +15x³ – 9x² + 3 =

= 10x⁵ – 6x⁴ +15x³ – 7x² +3