Machinery's Handbook, 31st Edition
Polynomials
29
Examples : ( )( x x 6 1
2
2
x x x = ∙ + ∙ – ∙ – ∙ = ∙ – ∙ 2 3 ) x x
x 1 6 6 1
x x x = + − − = − − x x 1 6 6 5 6
) x x x − + )( + 2
− 3 534 53 24 2 512 15 8 10 2 3 5 2 4 ( ) − = − + − x x x x + ∙ + x x x ∙
x
( 3 2 4
2
2
1 − + x x 1 )( )
1 1 1 1 x = ∙ + ∙ – ∙ – ∙ x x x
1 1 1 = + − − = − x x x x
(
1
(the middle terms drop out)
) ( x x x x x + =+ +=+++=++ 1 1 1 1 2 1 2 2 2 )( x x )
x
(
In the first and third examples, the products of the outer and inner terms are like terms, so they are combined. The third example is a special product (in this case, the difference of squares). The fourth is an example of raising a polynomial to an exponent. Raising a Polynomial to an Exponent: Consider first ( a + b ) 2 , which is ( a + b )( a + b ), and this expands by FOIL to a 2 + ab + ab + b 2 = a 2 + 2 ab + b 2 . In general, an expansion like ( a + b ) 2 , ( a – b ) 2 , ( a + b ) 3 , ( a – b ) 3 is called a binomial expansion , ( a + b ) n . For example: ab ababab ab n n + = + + + + ( ) ( )( )( ) ( ) factors Caution : A common but serious mistake is distributing the exponent to each term in the parentheses: ab abab a abb + = + + = + + ( ) ( )( ) 2 2 2 2 NOT ≠ + a b 2 2 Factoring Polynomials.—The result of two or more polynomials being multiplied is a higher-degree polynomial. Factoring a polynomial breaks it into its lesser-degree fac- tors. This section explains how to factor a polynomial, a skill needed for solving equa- tions and graphing functions (see Graphs of Functions on page 35). The categories are: dividing by a common factor (i.e., distributive property in reverse); factoring by reverse FOIL (with leading coefficient of 1 and otherwise); factoring special products . Common Factors: Each term is divided by the greatest common factor (GCF) and writ- ten as shown: ab + ac = a ( b + c ) Examples of GCF factoring : 6 2 10 2 3 5 3 2 2 x x x x x x + − = + ( ) − GCF + − = + ( ) 1 2 2 mnmnmn mn mn − GCF − − + = − + 7 21 4 7 21 4 2 abc bc bc bca c ( ) GCF − Note : The variable in the GCF is the one with the lowest exponent in common to all terms. If two terms have a GCF, but the others do not, then that factor cannot be pulled out. And, if the leading coefficient is negative (third example), it is customary to factor out –1 in the GCF redundant. This is so the expression in parentheses has a positive leading coefficient, which makes it easier to factor further. Reverse FOIL of Form x 2 + bx + c : The basic technique is demonstrated for a second- degree trinomial with a leading coefficient of 1. The task is to factor x 2 + bx + c as ( x + □ ) ( x + □ ), using only integers {. . . ,–2,–1, 0, 1, 2, . . .} in the boxes. Example : Consider x 2 + 4 x + 3. To factor it as ( x + □ )( x + □ ), the integers must have a product of +3 (the last term of x 2 + 4 x + 3) and a sum of +4 (the middle term). 1 and 3 are correct, as verified by FOIL: x 2 + 4 x + 3 = ( x + 1)( x + 3). As long as the leading coefficient is 1, reverse FOIL works. If one or both operations in the trinomial are negative, the process is the same, but some trial and error may be needed, as shown in the examples below.
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