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# operations with polynomials

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