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Sect. 5.1 Polynomials and Polynomial Functions Definitions Polynomial Functions Terms Degree of terms and polynomials Evaluating Graphing Simplifying by Combining Like Terms Adding & Subtracting Polynomials 5.1 1 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 -11xy2 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 92xyz5 Binomial – two terms: a + b or 7x5 – 44 Trinomial – three terms: x2 + 6x + 9 or a + b – 55c 5.1 2 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: 12 2 x 7 x x 2 y 2 5 x 3x y 7 xy 12 7 x x 2 x 3 3x y 5 x 7 xy y 2 3 2 4 2 3 5.1 2 3 2 4 3 2 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?). 5.1 4 Polynomial Functions Equations in one variable: f(x) = 2x – 3 (straight line) g(x) = x2 – 5x – 6 (parabola) h(x) = 3x3 + 4x2 – 2x + 5 Evaluate by substitution: f(5) = 2(5) – 3 = 10 – 3 = 7 g(-2) = (-2)2 – 5(-2) – 6 = 4 + 10 – 6 = 8 h(-1) = 3(-1)3 + 4(-1)2 – 2(-1) + 5 = -3+ 4+2+5= 8 5.1 5 Opposites of Monomials The opposite of a monomial has a different sign The opposite of 36 is -36 The opposite of -4x2 is 4x2 Monomial:Opposite: -2 5y ¾y5 -x3 0 2 -5y -¾y5 x3 0 5.1 6 Writing Any Polynomial as a Sum -5x2 – x is the same as -5x2 + (-x) Replace subtraction with addition: Keep the negative sign with the monomial 4x5 – 2x6 – 4x + 7 is 4x5 + (-2x6) + (-4x) + 7 You try it: -y4 + 3y3 – 11y2 – 129 -y4 + 3y3 + (-11y2) + (-129) 5.1 7 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 + 6x2 – 2x + 11xy Like terms: 3x, -x, -2x Also: y, -4y You try: 6x2 – 2x2 – 3 + x2 – 11 Like terms: 6x2, -2x2, x2 Also: -3, -11 5.1 8 Collecting Like Terms (simplifying) The numeric factor in a term is its coefficient. 3x + y – x – 4y + 6x2 – 2x 3 1 -1 -4 6 -2 You can simplify a polynomial by collecting like terms, summing their coefficients Let’s try: 6x2 – 2x2 – 3 + x2 – 11 Sum of: 6x2 + -2x2 + x2 is 5x2 Sum of: -3 + -11 is -14 Simplified polynomial is: 5x2 – 14 5.1 9 Collection Practice 2x3 – 6x3 = -4x3 5x2 + 7 + 4x4 + 2x2 – 11 – 2x4 = 2x4 + 7x2 – 4 4x3 – 4x3 = 0 5y2 – 8y5 + 8y5 = 5y2 ¾x3 + 4x2 – x3 + 7 = -¼x3 + 4x2 + 7 -3p7 – 5p7 – p7 = -9p7 5.1 10 Missing Terms x3 – 5 is missing terms of x2 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: x3 + 0x2 + 0x – 5 With space left: x3 –5 5.1 11 Adding 2 Polynomials - Horizontal To add polynomials, remove parentheses and combine like terms. (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 = 9x – 3 (5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2 – x – 6 = 5x2 – 4x – 4 This is called the horizontal method because you work left to right on the same “line” 5.1 12 Adding 2 Polynomials - Vertical 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 (5x2 – 3x + 2) + (-x – 6) = This is called the vertical method because you work from top to 5.1 13 bottom. More than 2 polynomials can be added at the same time. 5x2 – 3x + 2 + –x–6 5x2 – 4x – 4 Opposites of Polynomials The opposite of a polynomial has a reversed sign for each monomial The opposite of The opposite of Polynomial: -x + 2 3z – 5y ¾y5 + y5 – ¼y5 -(x3 – 5) y + 36 is -y – 36 -4x2 + 2x – 4 is 4x2 – 2x + 4 Opposite: x–2 -3z + 5y -¾y5 – y5 + ¼y5 x3 – 5 5.1 14 Subtracting Polynomials To subtract polynomials, add the opposite of the second polynomial. (7x3 + 2x + 4) – (5x3 – 4) add the opposite! (7x3 + 2x + 4) + (-5x3 + 4) Use either horizontal or vertical addition. Sometimes the problem is posed as subtraction: x2 + 5x +6 make it addition x2 + 5x +6 - (x2 + 2x) _ of the opposite -x2 – 2x__ 3x +6 5.1 15 Subtractions for You (5 x 2 4) (2 x 2 3x 1) 5x 2 4 2 x 2 3x 1 7 x 2 3x 5 (4 x 2 y 6 x 3 y 2 x 2 y 2 ) (4 x 2 y x 3y 2 3x 2 y 3 8 x 2 y 2 ) 4 x 2 y 6 x3 y 2 x2 y 2 4 x 2 y x 3y 2 3x 2 y 3 8 x 2 y 2 7 x 3y 2 3x 2 y 3 9 x 2 y 2 5.1 16 What Next? Section 5.2 – Multiplication of Polynomials 5.1 17