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Algebra Concepts Test #5 Review Page 1 of 4 Section 5.1: The Product and Power Rules for Exponents Rule Name Product Rule Rule a m a n a mn a a mn 3 ab a mb m 2 p m n Power Rules Example 6 2 6 4 6 2 4 66 m 2 4 m 5 32(4) 38 25 p 5 2 am a m b b 2 5 5 2 3 3 Section 5.2: Integer Exponents and the Quotient Rule Big Idea: There are shortcut formulas to doing calculations with exponents. These rules help us simplify expressions to make equation solving easier. Rules for Exponents: For positive integers m and n and any nonzero real numbers a and b: Rule Name Zero Exponent Definition Negative Exponent Definition Rule a0 1 am bn m a b b a m a a mn n a Quotient Rule 1 1 an bn m a an Negative to Positive Rules 4 Example 170 = 1; (-4)0 = 1 1 1 32 2 3 9 5 2 3 4 5 2 4 3 a 1b 2 3a 5a b 2 3 5 2 5 m 3 4 2 b 4 1 1 5a 1 2 2 a b 3a 3 4 2 b 2 2 4 2 3 5 2 1 4 a b 5 a 6b8 32 a 6b 10 42 a 2b 4 a 6 a 6b10 52 b8 32 42 a14b14 2 2 8 3 5 b 42 a14b 6 2 2 3 5 5 2 3 3 2 8 2 283 25 3 2 b Algebra Concepts Test #5 Review Page 2 of 4 Section 5.3: An Application of Exponents: Scientific Notation Examples: 4,280 = 4.28 103 93,000,000 = 9.3 107 753,658,554 7.54 109 128,000,000,000,000 = 1.28 1014 2.2 102 = 220 4.56 108 = 456,000,000 0.032 = 3.2 10-2 0.000 259 828 2.60 10-4 0.000 000 089 = 8.9 10-8 0.000 000 000 000 001 = 1 10-15 7.2 10-3 = 0.007 2 8.975 10-6 = 0.000 008 975 1. Converting from numbers to scientific notation: Move the decimal until it is just behind the first nonzero digit. The number of times you moved the decimal is the exponent on 10. The sign of the exponent will be positive fro big numbers, negative for small decimals. 2. Converting from scientific notation to numbers: For positive exponents, move the decimal to the right a number of times equal to the exponent on 10. For negative exponents, move the decimal to the left a number of times equal to the exponent on 10. 3. Scientific Notation on your calculator: Use the EE or EXP button: 3.810-6 = 3.8 EE -6 4. Calculating with scientific notation: Use your calculator to punch it in, or use the rules of exponents to work it out: 9.5 10 3.6 10 9.5 2 2 5 103 1 104 3.6 10 1 5 103 9.52 3.6 104 10 1 5 103 64.98 104 ( 1)( 3) 2 64.98 102 6.498 10 102 6.48 101 Algebra Concepts Test #5 Review Page 3 of 4 Section 5.4: Adding and Subtracting Polynomials; Graphing Simple Polynomials Polynomial vocabulary: A polynomial in x is a sum of finite terms of the form axn. Example: 4x3 + 6x2 – 5x – 8 Notice that the polynomial is written in descending powers of the variable. The coefficient of any term is the number part The degree of a term is the exponent on the variable. The degree of a polynomial is the greatest exponent. A polynomial with one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Polynomials with more terms aren’t usually given specific names. When adding or subtracting polynomials, you can only combine like terms. Example of evaluating polynomials: if the function P(x) = 5x2 + 3x – 7, then P(2) = 5(2)2 + 3(2) – 7 = 5(4) + 6 – 7 = 19 The graph of a polynomial of degree 2 (i.e., a quadratic polynomial) always looks like a parabola. 1 Here is a graph of equation y x 2 x : 64 Algebra Concepts Test #5 Review Page 4 of 4 Section 5.5: Multiplying Polynomials Multiplying two monomials: Use the rules of exponents to combine common bases. (-8p3q6)(-9p2q2) = (-8)(-9)(p3p2)(q6q2) = 728p5q8 Multiplying a monomial and a polynomial: Use the distributive property. -5x3(2x3 + 4x2 – 7x – 9) = -10x6 – 8x5 + 35x4 + 45 Multiplying two binomials: Use the FOIL acronym: multiply FIRST terms, then OUTSIDE terms, then INSIDE terms, then LAST terms. This method ONLY WORKS FOR BINOMIALS! Multiplying two polynomials: Multiply every term of the first polynomial with every term of the second polynomial, and then add up the products. There are three techniques for doing this: 1. Technique #1 for multiplying two polynomials: Write out every combination of products, then combine like terms. 2. Technique #2 for multiplying two polynomials: Write out the multiplication vertically, then carry it out like a long-hand numerical multiplication problem. 3. Technique #3 for multiplying two polynomials: Write out the polynomials on a grid, multiply out each pair of monomials, then add like terms. Section 5.6: Special Products (of Polynomials) 2 x y x 2 2 xy y 2 x y x 2 2 xy y 2 x y x y x2 y 2 2