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Arithmetic Of Polynomials I. Return Midterm II II. Homework III. Definition of a Polynomial (Vocabulary) Definition: A Monomial is a number times a power of x: ax2 Examples: 3x2, 1/2 x7, and 8 are all monomials. Definition: A polynomial is a sum or difference of monomials Examples: 4x5 - 3x2 - 1, 4x2, 2 are all polynomials. The degree of a polynomial is the largest power of x, the leading coefficient is the number in front of the term with the highest power of x, and the constant term is the number without any x's. Example: For the polynomial 4x5 - 3x2 - 1, the degree is 5, the leading coefficient is 4 and the constant term is -1. Notation: When we write P(x) = 3x3 - 2x2 +1 we say "P of x" To evaluate P(1), we find 3(1)3 - 2(1)2 + 1 = 2. IV. Addition and Subtraction of Polynomials To add or subtract polynomials, we just collect like terms: Example: Let P(x) = x2 + 3x + 5 and Q(x) = 4x3 - 2x2 + 3x - 2 Then P(x) - Q(x) = (x2 + 3x + 5) - (4x3 - 2x2 + 3x - 2) = x2 + 3x + 5 - 4x3 + 2x2 - 3x + 2 = -4x3 + 3x2 + 7 Exercise: Let P(x) = 3x2 + 4x - 2 and Q(x) = 5x2 - 3x - 5 Find P(x) + Q(x) V. FOIL Consider the multiplication of the following two first degree polynomials: (x + 3)(x + 4) = (x + 3)x + (x + 3)4 = x2 + 3x + 4x + 12 = x2 + 7x + 12 Since this type of multiplication occurs so frequently, we have a systematic approach called FOIL- Firsts, Outers, Inners, Lasts. That is we multiply the first terms the outer terms, the inner terms, and the last terms add the four results together. Examples: A) (x + 2)(x + 5) = x2 + 5x + 2x + 10 = x2 + 7x + 10 B) (3x - 4)(5x + 2) = 15x2 + 6x - 20x - 8 Exercises: A) (x - 2)(3x + 1) B) (5x + 4)(3x + 2) C) (3x - y)(2x + 3y) D) (x + y)(x - y) E) (x + y)(x + y) F) (x - y)(x - y) We will note the special products D, E and F as difference of squares, perfect square of sum, and perfect square of difference. Examples: A) (3 - x)(3 + x) = 9 - x2 B) (x + 3)2 = x2 + 6x + 9 C) (2x - 4)2 = 4x2 - 16x + 16 D) (x + 2)3 = (x + 2)2(x + 2) = (x2 + 4x + 4)(x + 2) = (x2 + 4x + 4)x + (x2 + 4x + 4)2 = x3 + 4x2 + 4x + 2x2 + 8x + 8 = x3 + 6x2 + 12x + 8. VI. General Polynomial Multiplication When the polynomials have more than two terms, we must use the distributive property as follows: Example (x3 -3x +1)(x - 3) = (x3 -3x +1)(x) + (x3 -3x +1)(-3) = x4 - 3x2 + x - 3x3 + 9x - 3 = x4 - 3x3 -3x2 + 10x - 3
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