Machinery's Handbook, 31st Edition
28
Polynomials Polynomials Polynomials in a single variable are expressions of the form a x a x a x n n + + − 1
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a x a
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a x a , where n is a non-negative integer , and the coefficients a n , a n –1 , . . . , a 0 are real numbers (the subscripts are simply labels that correspond to the variables ( x n , x n –1 , . . .). n is the degree (or order ) of the polynomial; a n is the leading coefficient ; a 0 is the constant coefficient . A first-degree polynomial has n = 1; a second-degree polynomial has n = 2, and so on. Example : x 3 1 6 + is degree 1, a 1 = 1 3 is the leading coefficient, a 0 = 1 6 is the constant coefficient. Example : 4 7 2 x − is degree 2, a 2 = 4 is the leading coefficient, a 0 =–7 is the constant coefficient. Example : x x x 3 2 5 1 3 + + − is degree 3, and the coefficients are a 3 = 1, a 2 = 5, a 1 =–1, and a 0 = 1 3 . + + + 1 1 0 A monomial is a single-term polynomial. For example, 8 x 5 is degree 5 with coefficient 8. A constant is a polynomial, by the definition. Its degree is 0. For example, 19 is a 0-degree polynomial, since it can be written as 19 x 0 . A binomial is a polynomial with two terms, such as x + 9 and 5 – x 2 . A trinomial has three terms, such as, x 2 + x – 6. Finally, second-degree polynomials are called quadratic polynomials. They are im- portant because they model so many processes in engineering and other technical and scientific fields. Operations on Polynomials.—Polynomials can be added, subtracted, or multiplied (“expanded”), with the result being another polynomial. If polynomials are divided, the result is a rational expression . Polynomials also may be factored . That is, they may be written as a product of lower-degree polynomials. Combining (Adding and Subtracting) Polynomials: Two or more polynomials are added by combining like terms. For example: x x x x x x x x x x x 4 3 2 3 2 4 3 2 6 3 5 11 12 28 7 15 4 17 + − + + + + − − ( ) − + ( ) = − − Multiplying (Expanding) Polynomials: Taking the product of two or more polynomials relies on the distributive property of multiplication over addition (or subtraction): a ( b + c ) = ab + ac, where the rules of exponents are followed (see Properties of Monomials and Exponents on page 27). Examples of simple distributive case : 2 8 3 6 2 8 2 3 2 6 16 6 12 4 2 1 4 1 2 5 3 x x x x x x x x x + − = + ( ) = − − + + ∙ + ∙ ∙ Note : This technique also is used for multiplying other algebraic expressions, such as radical (root) and rational expressions. Such terms are not polynomials, since their expo- nents are other than non-negative integers. Examples of Distributive Property for Root and Rational Expressions : 2 7 3 14 6 14 6 1 2 1 2 1 1 2 3 2 1 2 x x x x x x / / / / / + = + = + ( ) + 1 3 2 1 3 1 2 1 1 2 1 0 2 7 3 7 3 7 3 7 3 x x x x x x x x x x x x − ( ) = = = − + − + − + + − + − + − + “FOIL” is a version of the distributive property in which two binomials are multiplied. FOIL stands for F irst O uter I nner L ast, the order in which terms are multiplied. The first is ac ; outer is ad , inner is bc , and last is bd . FOIL multiplication: ( a + b )( c + d ) = ac + ad + bc + bd
x a x +
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