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1. –2 4 2. (–2) 4 3. x – 2(3x – 1) 4. 3(y 2 + 6y) –5x + 2 Simplify each expression. –16 16 3y 2 + 18y

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A monomial is a number or a product of numbers and variables with whole number exponents. A polynomial is a monomial or a sum or difference of monomials. Each monomial in a polynomial is a term. Because a monomial has only one term, it is the simplest type of polynomial. Polynomials have no variables in denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents. Polynomials: 3x43x4 2z 12 + 9z 3 1 2 a7a7 0.15x 101 3t 2 – t 3 Not polynomials: 3x3x |2b 3 – 6b| 8 5y25y2 m 0.75 – m The degree of a monomial is the sum of the exponents of the variables. 1 2

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Identify the degree of each monomial. Example 1: Identifying the Degree of a Monomial A. z 6 Identify the exponent. B. 5.6 The degree is 6. z6z6 5.6 = 5.6x 0 Identify the exponent. The degree is 0. C. 8xy 3 Add the exponents. D. a 2 bc 3 The degree is 4. 8x1y38x1y3 a2b1c3a2b1c3 Add the exponents. The degree is 6.

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An degree of a polynomial is given by the term with the greatest degree. A polynomial with one variable is in standard form when its terms are written in descending order by degree. So, in standard form, the degree of the first term indicates the degree of the polynomial, and the leading coefficient is the coefficient of the first term.

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A polynomial can be classified by its number of terms. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial. A polynomial can also be classified by its degree.

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Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. A. 3 – 5x 2 + 4xB. 3x 2 – 4 + 8x 4 –5x 2 + 4x + 3 Write terms in descending order by degree. Leading coefficient: –5 Terms: 3 Name: quadratic trinomial Degree: 2 8x 4 + 3x 2 – 4 Write terms in descending order by degree. Leading coefficient: 8 Terms: 3 Name: quartic trinomial Degree: 4

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Example 4: Work Application The cost of manufacturing a certain product can be approximated by f(x) = 3x 3 – 18x + 45, where x is the number of units of the product in hundreds. Evaluate f(0) and f(200) and describe what the values represent. f(0) represents the initial cost before manufacturing any products. f(200) represents the cost of manufacturing 20,000 units of the products. f(0) = 3(0) 3 – 18(0) + 45 = 45 f(200) = 3(200) 3 – 18(200) + 45 = 23,996,445

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Find each product. Example 1: Multiplying a Monomial and a Polynomial A. 4y 2 (y 2 + 3) Distribute. B. fg(f 4 + 2f 3 g – 3f 2 g 2 + fg 3 ) 4y 2 y 2 + 4y 2 3 4y 2 (y 2 + 3) Multiply. 4y 4 + 12y 2 fg(f 4 + 2f 3 g – 3f 2 g 2 + fg 3 ) Distribute. Multiply. fg f 4 + fg 2f 3 g – fg 3f 2 g 2 + fg fg 3 f 5 g + 2f 4 g 2 – 3f 3 g 3 + f 2 g 4

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To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first. Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

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(y 2 – 7y + 5)(y 2 – y – 3) Find the product. Multiply each term of one polynomial by each term of the other. Use a table to organize the products. y4y4 –y3–y3 –3y 2 –7y 3 7y27y2 21y 5y25y2 –5y–15 y 2 –y –3 y2y2 –7y 5 The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product. y 4 + (–7y 3 – y 3 ) + (5y 2 + 7y 2 – 3y 2 ) + (–5y + 21y) – 15 y 4 – 8y 3 + 9y 2 + 16y – 15

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Find the product. (3b – 2c)(3b 2 – bc – 2c 2 ) 3b(3b 2 ) + 3b(–2c 2 ) + 3b(–bc) – 2c(3b 2 ) – 2c(–2c 2 ) – 2c(–bc) Multiply horizontally. 9b 3 – 6bc 2 – 3b 2 c – 6b 2 c + 4c 3 + 2bc 2 9b 3 – 9b 2 c – 4bc 2 + 4c 3 Write polynomials in standard form. Distribute 3b and then –2c. Multiply. Add exponents. Combine like terms. (3b – 2c)(3b 2 – 2c 2 – bc)

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Find the product. (x + 4) 4 (x + 4)(x + 4)(x + 4)(x + 4) Write in expanded form. Multiply the last two binomial factors. (x + 4)(x + 4)(x 2 + 8x + 16) Multiply the first two binomial factors. (x 2 + 8x + 16)(x 2 + 8x + 16) x 2 (x 2 ) + x 2 (8x) + x 2 (16) + 8x(x 2 ) + 8x(8x) + 8x(16) + 16(x 2 ) + 16(8x) + 16(16) Distribute x 2 and then 8x and then 16. Multiply. x 4 + 8x 3 + 16x 2 + 8x 3 + 64x 2 + 128x + 16x 2 + 128x + 256 Combine like terms. x 4 + 16x 3 + 96x 2 + 256x + 256

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PAGE 333 #15-23 ODD AND PAGE 341 # 17-23ODD AND 29-35ODD, 43, 47, 55 HW 2

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