Section 5.3 Polynomials and Polynomial Functions

1. Section 5.3 Polynomials and Polynomial Functions Definitions Terms Degree of terms and polynomials Polynomial Functions Evaluating Graphing Simplifying by Combining Like Terms Adding & Subtracting Polynomials

2. Definitions

3. Definitions

4. DEGREE Of a Monomial: One variable – its degree is the variable’s exponent Several variables – the degree is the sum of their exponents Non-zero constant – the degree is 0 The constant 0 has an undefined degree Of a Polynomial: Its degree is the same as the degree of the term in the polynomial with largest degree.

5. Arranging Terms in a Standard Order Ascending Order – Constant term leads off Descending Order – Variable with largest exponent leads off

6. Polynomials as 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

7. Evaluating Polynomials

8. 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 2 5y -5y ¾y5 -¾y5 -x3 x3 0 0

9. 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)

10. Identifying Like Terms When two different terms in a polynomial have the exactly the same variables raised to exactly the same powers, we call them like terms. 3x + y – x – 4y + 6x2 – 2x Like terms: 3x, -x, -2x Also: y, -4y You try: 6x2 – 2x2 – 3 + x2 – 11 Like terms: 6x2, -2x2, x2 Also: -3, -11

11. Collecting Like Terms 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 + 1x2 is 5x2 Sum of: -3 + -11 is -14 Simplified polynomial is: 5x2 – 14

12. 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

13. Missing Terms x3 – 5 is missing terms of x2 and x So what! Leaving space for missing terms will help you when you add, subtract, multiply and divide polynomials You can write the expression above in 2 ways: With 0 coefficients: x3 + 0x2 + 0x – 5 With space left: x3 – 5

14. Adding 2 PolynomialsThe Horizontal Method To add polynomials, remove parentheses, then 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”

15. Adding 2 PolynomialsThe Vertical Method To add polynomials vertically, remove parentheses, put one over the other lining up like terms, then add the terms (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2  Add the matching columns 9x – 3 (5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2  + – x – 6 5x2 – 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

16. Opposites of Polynomials The opposite of a polynomial has a reversed sign for each monomial The opposite of y + 36 is -y – 36 The opposite of -4x2 + 2x – 4 is 4x2 – 2x + 4 Polynomial: Opposite: -x + 2 x – 2 3z – 5y -3z + 5y ¾y5 + y5 – ¼y5 -¾y5 – y5 + ¼y5 -(x3 – 5) x3 – 5

17. 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) = 2x3 + 2x + 8 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

18. Examples Perform the indicated operation

19. What Next? Present Section 5.4Multiplying Polynomials