M1410P03 - (Preliminaries Basic Algebra P.30 TOPIC 5...

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(Preliminaries: Basic Algebra) P.30 TOPIC 5: RATIONAL AND ALGEBRAIC EXPRESSIONS (See Section A.4 .) A rational expression in x can be expressed in the form polynomial in x nonzero polynomial in x . Examples: 1 x , 5 x 3 1 x 2 + 7 x 2 , x 7 + x which equals x 7 + x 1 Observe in the second example that irrational numbers such as 2 are permissible. The last example correctly suggests that all polynomials are rational expressions. An algebraic expression in x looks like a rational expression, except that radicals and exponents that are noninteger rational numbers such as 5 7 are allowed even when x appears in a radicand or in a base (but not in an exponent). Examples: x , x 3 + 7 x 5/7 x x + 5 3 + 4 All rational expressions are algebraic.
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(Preliminaries: Basic Algebra) P.31 Venn diagram for standard symbolic mathematical expressions:
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(Preliminaries: Basic Algebra) P.32 TOPIC 6: FACTORING WITH “WEIRD” EXPONENTS: NEGATIVE AND FRACTIONAL EXPONENTS; QUADRATIC FORMS AND SUBSTITUTIONS (See Section A.4: p.A41, and Section A.7: pp.A72-A73 .) These manipulations come up throughout Calculus, especially Calculus I ( Math 150 at Mesa ). Basic Factoring Example We factor 8 x + 6 as 2 4 x + 3 ( ) . Observe that, when we factor out a 2, we divide both terms in 8 x + 6 by 2. Basic Factoring Example We factor x 5 + x 3 as x 3 x 2 + 1 ( ) . Observe that we factor out the power of x with the lowest exponent. Warning: All negative exponents are lesser than all positive exponents. See the Example below. When we divide x 5 by x 3 , we subtract the exponents in that order and get x 2 . We can also think, “ x 3 times what gives us x 5 ?” We can also think, “We’re factoring x 3 out of x 5 . 5 takeaway 3 is 2.” Example Factor x 7 + x 4 2 x 1 completely over the integers. Observe that 7 is the lowest exponent on x . We will factor out x 7 and subtract 7 from each of the exponents. x 7 + x 4 2 x 1 = x 7 1 + x 4 − − 7 ( ) 2 x 1 − − 7 ( ) ( ) = x 7 1 + x 4 + 7 2 x 1 + 7 ( ) = x 7 1 + x 3 2 x 6 ( )
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(Preliminaries: Basic Algebra) P.33 Observe that 1 + x 3 2 x 6 ( ) has no negative exponents. We can actually factor x 7 1 + x 3 2 x 6 ( ) further over the integers. We will first factor out a 1 factor so that the leading coefficient of our trinomial factor is positive. Then, we will rewrite the trinomial in descending powers. The trinomial will then be easier to factor. = x 7 1 x 3 + 2 x 6 ( ) = x 7 2 x 6 x 3 1 ( ) The trinomial 2 x 6 x 3 1 is in quadratic form, because the exponent on x in the first term is twice that in the second term (6 is twice 3), and the third term is a nonzero constant. This means that the techniques for factoring quadratic trinomials (see Topic 4 ) may be useful here.
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