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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 1 § 5.2 More Work with Exponents and Scientific Notation

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 2 The Power Rule and Power of a Product or Quotient Rule for Exponents If a and b are real numbers and m and n are integers, then The Power Rule (ab) n = a n · b n Power Rule(a m ) n = a mn Power of a Product Power of a Quotient

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 3 Simplify each of the following expressions. (2 3 ) 3 = 2 9 = 512 (x4)2(x4)2 = x 8 = 2 3·3 = x 4·2 The Power Rule Example: = 5 3 · (x 2 ) 3 · y 3 = 125x 6 y 3 (5x 2 y) 3

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 4 If m and n are integers and a and b are real numbers, then: Product Rule for exponents a m · a n = a m+n Power Rule for exponents (a m ) n = a mn Power of a Product (ab) n = a n · b n Power of a Quotient Quotient Rule for exponents Zero exponent a 0 = 1, a 0 Negative exponent Summary of Exponent Rules

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 5 Simplify by writing the following expression with positive exponents or calculating. Power of a quotient rulePower of a product rule Quotient rule Simplifying Expressions Power rule Product rule Negative exponents

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 6 Operations with Scientific Notation Example Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Perform the following operations. = (7.3 · 8.1) (10 2 · 10 5 ) = 59.13 10 3 = 59,130 (7.3 10 2 )(8.1 10 5 ) 1) 2)

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 7 Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Perform the following operations. = (7.3 · 8.1) (10 -2 ·10 5 ) = 59.13 10 3 = 59,130 (7.3 10 2 )(8.1 10 5 ) 1) 2) Operations with Scientific Notation Example

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 8 § 5.3 Polynomials and Polynomial Functions

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 9 Polynomial Vocabulary Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 10 In the polynomial 7x 5 + x 2 y 2 – 4xy + 7 There are 4 terms: 7x 5, x 2 y 2, -4xy and 7. The coefficient of term 7x 5 is 7, of term x 2 y 2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term. Polynomial Vocabulary

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 11 Monomial is a polynomial with one term. Binomial is a polynomial with two terms. Trinomial is a polynomial with three terms. Types of Polynomials

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 12 Degree of a term To find the degree, take the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a 4 b 3 c is 8 (remember that c can be written as c 1 ). Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9x 3 – 4x 2 + 7 is 3. Degrees

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 13 Like terms are terms that contain exactly the same variables raised to exactly the same powers. Combine like terms to simplify. x 2 y + xy – y + 10x 2 y – 2y + xy Only like terms can be combined through addition and subtraction. Warning! 11x 2 y + 2xy – 3y= (1 + 10)x 2 y + (1 + 1)xy + (– 1 – 2)y = = x 2 y + 10x 2 y + xy + xy – y – 2y (Like terms are grouped together) Combining Like Terms Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 14 Adding Polynomials To add polynomials, combine all the like terms. Adding Polynomials Add. (3x – 8) + (4x 2 – 3x +3) = 4x 2 + 3x – 3x – 8 + 3 = 4x 2 – 5 = 3x – 8 + 4x 2 – 3x + 3 Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 15 Subtracting Polynomials To subtract polynomials, add its opposite. Subtracting Polynomials Example: Subtract. = 3a 2 – 6a + 11 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4= y + 8 (– a 2 + 1) – (a 2 – 3) + (5a 2 – 6a + 7) = – a 2 + 1 – a 2 + 3 + 5a 2 – 6a + 7 = – a 2 – a 2 + 5a 2 – 6a + 1 + 3 + 7

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 16 In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression. You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically. Adding and Subtracting Polynomials

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 17 § 5.4 Multiplying Polynomials

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 18 Multiplying Two Polynomials To multiply any two polynomials, use the distributive property and multiply each term of one polynomial by each term of the other polynomial. Then combine like terms. Multiplying Polynomials

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 19 Multiply each of the following. 1) (3x 2 )( – 2x) = (3)( – 2)(x 2 · x) = – 6x 3 2) (4x 2 )(3x 2 – 2x + 5) = (4x 2 )(3x 2 ) – (4x 2 )(2x) + (4x 2 )(5) (Distributive property) = 12x 4 – 8x 3 + 20x 2 (Multiply the monomials) 3) (2x – 4)(7x + 5)= 2x(7x + 5) – 4(7x + 5) = 14x 2 + 10x – 28x – 20 = 14x 2 – 18x – 20 Multiplying Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 20 Multiply (3x + 4) 2 Remember that a 2 = a · a, so (3x + 4) 2 = (3x + 4)(3x + 4). (3x + 4) 2 = (3x + 4)(3x + 4)= (3x)(3x + 4) + 4(3x + 4) = 9x 2 + 12x + 12x + 16 = 9x 2 + 24x + 16 Multiplying Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 21 Multiply (a + 2)(a 3 – 3a 2 + 7). (a + 2)(a 3 – 3a 2 + 7) = a(a 3 – 3a 2 + 7) + 2(a 3 – 3a 2 + 7) = a 4 – 3a 3 + 7a + 2a 3 – 6a 2 + 14 = a 4 – a 3 – 6a 2 + 7a + 14 Multiplying Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 22 Multiply (3x – 7y)(7x + 2y) (3x – 7y)(7x + 2y)= (3x)(7x + 2y) – 7y(7x + 2y) = 21x 2 + 6xy – 49xy + 14y 2 = 21x 2 – 43xy + 14y 2 Multiplying Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 23 Multiply (5x – 2z) 2 (5x – 2z) 2 = (5x – 2z)(5x – 2z)= (5x)(5x – 2z) – 2z(5x – 2z) = 25x 2 – 10xz – 10xz + 4z 2 = 25x 2 – 20xz + 4z 2 Multiplying Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 24 Multiply (2x 2 + x – 1)(x 2 + 3x + 4) (2x 2 + x – 1)(x 2 + 3x + 4) = (2x 2 )(x 2 + 3x + 4) + x(x 2 + 3x + 4) – 1(x 2 + 3x + 4) = 2x 4 + 6x 3 + 8x 2 + x 3 + 3x 2 + 4x – x 2 – 3x – 4 = 2x 4 + 7x 3 + 10x 2 + x – 4 Multiplying Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 25 You can also use a vertical format in arranging the polynomials to be multiplied. In this case, as each term of one polynomial is multiplied by a term of the other polynomial, the partial products are aligned so that like terms are together. This can make it easier to find and combine like terms. Multiplying Polynomials

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 26 Multiply (2x – 4)(7x + 5) (2x – 4)(7x + 5) = = 14x 2 + 10x – 28x – 20 2x(7x)+ 2x(5)– 4(7x)– 4(5) = 14x 2 – 18x – 20 We multiplied these same two binomials together in the previous section, using a different technique, but arrived at the same product. Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 27 Square of a Binomial (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 – 2ab + b 2 Product of the Sum and Difference of Two Terms (a + b)(a – b) = a 2 – b 2 Special Products

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 28 Although you will arrive at the same results for the special products by using the techniques of this section or last section, memorizing these products can save you some time in multiplying polynomials. Special Products

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 29 We can use function notation to represent polynomials. For example, P(x) = 2x 3 – 3x + 4. Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Find the value P( 2) = 2x 3 – 3x + 4. = 2( 2) 3 – 3( 2) + 4P(2)P(2) = 2( 8) + 6 + 4 = 6 Evaluating Polynomials Example:

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Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed 30 Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values. If f(x) = 2x 2 + 3x – 4, find f(a + 3). We replace the variable x with a + 3 in the polynomial function. f(a + 3) = 2(a + 3) 2 + 3(a + 3) – 4 = 2(a 2 + 6a + 9) + 3a + 9 – 4 = 2a 2 + 12a + 18 + 3a + 9 – 4 = 2a 2 + 15a + 23 Evaluating Polynomials Example:

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