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Sect. 5.1 Polynomials and Polynomial Functions Definitions Terms Degree of terms and polynomials Polynomial Functions Evaluating Graphing Simplifying by Combining Like Terms Adding & Subtracting Polynomials 5.11

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Definitions An algebraic term is a number or a product of a number and a variable (or variables) raised to a positive power. Examples: 7x or -11xy 2 or 192 or z A constant term contains only a number A variable term contains at least one variable and has a numeric part and a variable part A polynomial expression is: one or more terms separated by addition or subtraction; any exponents must be whole numbers; no variable in any denominator. Monomial – one term: -47 or 7x or 92xyz 5 Binomial – two terms: a + b or 7x 5 – 44 Trinomial – three terms: x 2 + 6x + 9 or a + b – 55c 5.12

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Ordering a Polynomial’s Terms If there are multiple variables, one must be specified Descending Order – Variable with largest exponent leads off (this is the Standard) Ascending Order – Constant term leads off Arrange in ascending order: Arrange in descending order of x: 5.13

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DEGREE Of a Monomial: (a Single Term) Several variables – the degree is the sum of their exponents One variable – its degree is the variable’s exponent Non-zero constant – the degree is 0 The constant 0 has an undefined degree Of a Polynomial: (2 or more terms added or subtracted) Its degree is the same as the degree of the term in the polynomial with largest degree (leading term?). 5.14

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Polynomial Functions Equations in one variable: f(x) = 2x – 3 (straight line) g(x) = x 2 – 5x – 6 (parabola) h(x) = 3x 3 + 4x 2 – 2x + 5 Evaluate by substitution: f(5) = 2(5) – 3 = 10 – 3 = 7 g(-2) = (-2) 2 – 5(-2) – 6 = – 6 = 8 h(-1) = 3(-1) 3 + 4(-1) 2 – 2(-1) + 5 = =

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Opposites of Monomials The opposite of a monomial has a different sign The opposite of 36 is -36 The opposite of -4x 2 is 4x 2 Monomial:Opposite: -22 5y-5y ¾y5-¾y5¾y5-¾y5 -x 3 x

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Writing Any Polynomial as a Sum -5x 2 – x is the same as -5x 2 + (-x) Replace subtraction with addition: Keep the negative sign with the monomial 4x 5 – 2x 6 – 4x + 7 is 4x 5 + (-2x 6 ) + (-4x) + 7 You try it: -y 4 + 3y 3 – 11y 2 – 129 -y 4 + 3y 3 + (-11y 2 ) + (-129) 5.17

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Identifying Like Terms When several terms in a polynomial have the same variable(s) raised to the same power(s), we call them like terms. 3x + y – x – 4y + 6x 2 – 2x + 11xy Like terms: 3x, -x, -2x Also: y, -4y You try: 6x 2 – 2x 2 – 3 + x 2 – 11 Like terms: 6x 2, -2x 2, x 2 Also: -3,

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Collecting Like Terms (simplifying) The numeric factor in a term is its coefficient. 3x + y – x – 4y + 6x 2 – 2x You can simplify a polynomial by collecting like terms, summing their coefficients Let’s try: 6x 2 – 2x 2 – 3 + x 2 – 11 Sum of: 6x x 2 + x 2 is 5x 2 Sum of: is -14 Simplified polynomial is: 5x 2 –

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2x 3 – 6x 3 = -4x 3 5x x 4 + 2x 2 – 11 – 2x 4 = 2x 4 + 7x 2 – 4 4x 3 – 4x 3 = 0 5y 2 – 8y 5 + 8y 5 = 5y 2 ¾x 3 + 4x 2 – x = -¼x 3 + 4x -3p 7 – 5p 7 – p 7 = -9p 7 Collection Practice 5.110

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Missing Terms x 3 – 5 is missing terms of x 2 and x So what! Leaving space for missing terms will help you when you start adding & subtracting polynomials Write the expression above in either of two ways: With 0 coefficients: x 3 + 0x 2 + 0x – 5 With space left: x 3 –

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To add polynomials, remove parentheses and combine like terms. (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 = 9x – 3 (5x 2 – 3x + 2) + (-x – 6) = 5x 2 – 3x + 2 – x – 6 = 5x 2 – 4x – 4 This is called the horizontal method because you work left to right on the same “line” Adding 2 Polynomials - Horizontal 5.112

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To add polynomials vertically, remove parentheses, put one over the other lining up like terms, add terms. (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 Add the matching columns 9x – 3 (5x 2 – 3x + 2) + (-x – 6) = 5x 2 – 3x – x – 6 5x 2 – 4x – 4 This is called the vertical method because you work from top to bottom. More than 2 polynomials can be added at the same time. Adding 2 Polynomials - Vertical 5.113

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The opposite of a polynomial has a reversed sign for each monomial The opposite of y + 36 is -y – 36 The opposite of -4x 2 + 2x – 4 is 4x 2 – 2x + 4 Polynomial:Opposite: -x + 2x – 2 3z – 5y-3z + 5y ¾y 5 + y 5 – ¼y 5 - ¾y 5 – y 5 + ¼y 5 -(x 3 – 5)x 3 – 5 Opposites of Polynomials 5.114

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Subtracting Polynomials To subtract polynomials, add the opposite of the second polynomial. (7x 3 + 2x + 4) – (5x 3 – 4) add the opposite! (7x 3 + 2x + 4) + (-5x 3 + 4) Use either horizontal or vertical addition. Sometimes the problem is posed as subtraction: x 2 + 5x +6 make it addition x 2 + 5x +6 - (x 2 + 2x) _ of the opposite -x 2 – 2x__ 3x

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Subtractions for You 5.116

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What Next? Section 5.2 – Multiplication of Polynomials Section

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