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

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

Exercise: Let A = x³ + 2x² - x + 7, and B = -2x² + 7. Calculate:

a) A + B

b) A - B

c) A · B

d) A · 2B

Solutions: a) x³ - x + 14; b) x³ + 8x² - x -14; c) -2x⁵ - 4x⁴ + 9x³ -7x + 49; d) -4x⁵ - 8x⁴ + 18x³ -14x + 98