Precalc0006to0010

Precalc0006to0010 - (Section 0.6: Polynomial, Rational, and...

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(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES • Be able to identify polynomial, rational, and algebraic expressions. • Understand terminology and notation for polynomials. PART A: DISCUSSION • In Chapters 1 and 2, we will discuss polynomial, rational, and algebraic functions, as well as their graphs. PART B: POLYNOMIALS Let n be a nonnegative integer. An n th -degree polynomial in x , written in descending powers of x , has the following general form : a n x n + a n ± 1 x n ± 1 + ... + a 1 x + a 0 , a n ² 0 () The coefficients , denoted by a 1 , a 2 , , a n , are typically assumed to be real numbers, though some theorems will require integers or rational numbers. a n , the leading coefficient , must be nonzero , although any of the other coefficients could be zero (i.e., their corresponding terms could be “missing”). a n x n is the leading term . a 0 is the constant term . It can be thought of as a 0 x 0 , where x 0 = 1 . • Because n is a nonnegative integer, all of the exponents on x indicated above must be nonnegative integers, as well. Each exponent is the degree of its corresponding term.
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(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.2 Example 1 (A Polynomial) 4 x 3 ± 5 2 x 2 + 1 is a 3 rd -degree polynomial in x with leading coefficient 4, leading term 4 x 3 , and constant term 1. The same would be true even if the terms were reordered: 1 ± 5 2 x 2 + 4 x 3 . The polynomial 4 x 3 ± 5 2 x 2 + 1 fits the form a n x n + a n ± 1 x n ± 1 + ... + a 1 x + a 0 , with degree n = 3 . It can be rewritten as: 4 x 3 ± 5 2 x 2 + 0 x + 1 , which fits the form a 3 x 3 + a 2 x 2 + a 1 x + a 0 , where the coefficients are: a 3 = 4 leading coefficient () a 2 = ± 5 2 a 1 = 0 a 0 = 1 constant term ² ³ ´ ´ ´ µ ´ ´ ´ § Example Set 2 (Constant Polynomials) 7 is a 0 th -degree polynomial. It can be thought of as 7 x 0 . 0 is a polynomial with no degree. §
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(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.3 PART C: CATEGORIZING POLYNOMIALS BY DEGREE Degree Type Examples 0 [Nonzero] Constant 7 1 Linear 3 x + 4 2 Quadratic 5 x 2 ± x + 1 3 Cubic x 3 + 4 x 4 Quartic x 4 ± ² 5 Quintic x 5 PART D: CATEGORIZING POLYNOMIALS BY NUMBER OF TERMS Number of Terms Type Examples 1 Monomial x 5 2 Binomial x 3 + 4 x 3 Trinomial 5 x 2 ± x + 1 PART E: SQUARING BINOMIALS Formulas for Squaring Binomials a + b () 2 = a 2 + 2 ab + b 2 a ± b 2 = a 2 ± 2 ab + b 2 WARNING 1 : When squaring binomials, don’t forget the “middle term” of the resulting Perfect Square Trinomial (PST) . For example, x + 3 2 = x 2 + 6 x + 9 . Observe that 6 x is twice the product of the terms x and 3: 6 x = 2 x 3 . The figure below implies that x + y 2 = x 2 + 2 xy + y 2 for x > 0 and y > 0 .
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(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.4 PART F: RATIONAL AND ALGEBRAIC EXPRESSIONS A rational expression in x can be expressed in the form: polynomial in x nonzero polynomial in x Example Set 3 (Rational Expressions) Examples of rational expressions include: a) 1 x .
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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Precalc0006to0010 - (Section 0.6: Polynomial, Rational, and...

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