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Understanding Polynomial Functions and Expressions

Learn the basics of polynomial functions and expressions, including degrees, leading coefficients, and modeling data. Explore multiplication of polynomials, factoring techniques, and solving polynomial equations.

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Understanding Polynomial Functions and Expressions

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  1. Chapter 5 Polynomial Expressions and Functions

  2. Ch 5 - Polynomial Functions Every linear function can be written as f(x) = ax +b and is an example of a polynomial function. However, polynomial functions of degree 2 or higher are nonlinear functions. To model nonlinear data we use polynomial functions of degree 2 or higher.

  3. 5.1 Polynomial Functions The following are examples of polynomial functions. f(x) = 3 Degree = 0 0 Constant f(x) = 5x – 3 Degree = 1 5 Linear f(x) = x2 – 2x –1 Degree = 2 1 Quadratic f(x) = 3x3 + 2x2 –6 Degree = 3 3 Cubic Polynomial Degree Leading Coefficient Type

  4. Polynomial Expressions A term is a number, a variable, or a product of numbers and variables raised to powers. Examples of terms include • 15, y, x4, 3x3 y, x-1/2 y-2 , and 6 x-1 y3 If the variables in a term have only nonnegative integer exponents, the term is called a monomial. Examples of monomials include • 4, 5y, x2, 5x 2 z4, - x y4 and 6xy4

  5. Monomials x x x x x x y y y Volume x3 Total Area xy + xy + xy = 3xy

  6. Modeling AIDS cases in the United StatesEx 12 ( Pg 327 ) 600 500 400 300 200 100 Aids Cases (Thousands) 0 4 6 8 10 12 14 Year ( 1984 1994 ) f(x) = 4.1x2 - 25x + 46 f(7) = 4.1(7) 2- 25.7 + 46 = 71.9 f(17) = 4.1 (17) 2 – 25.7 + 46 = 805.9

  7. Modeling heart rate of an athelet ( ex 13, pg 328 ) 250 200 150 100 50 Heart Rate (bpm) 0 1 2 3 4 5 6 7 8 Time (minutes ) P(t) = 1.875 t2– 30t + 200 Where 0 < t < 8 P(0) = 1.875(0) 2 – 30(0) + 200 = 200 (Initial Heart Rate) P(8) = 1.875(8) 2– 30(8 )+ 200= 80 beats per minute (After 8 minutes) The heart rate does not drop at a constant rate; rather, it drops rapidly at first and then gradually begins to level off

  8. A PC for all(No 120, Pg 332) Worldwide computer 200 150 100 50 0 1998 2000 2002 2004 Year f(x) = 0.7868 x2 + 12x + 79.5 x = 0 corresponds to 1997 x = 1 corresponds to 1998 and so on x = 6 corresponds to 2003 f(6) = 0.7868 x( 6) 2 + 12 x( 6 ) + 79.5 = 179.8248 = 180 million(approx)

  9. 5.2 Multiplications of Polynomials Using distributive properties Multiply 4 (5 + x) = 4.5 + 4. x = 20 + 4x 20 4x 4 • x • Area : 20 + 4x

  10. ….Cont Multiplying Binomials Multiply (x+ 1)(x + 3) • Geometrically • Symbolically a) Geometrically x + 1 1 x 3 x2 3x x + 3 x 3 Area: (x + 1)(x + 3) Area: x2 + 4x + 3 b) Symbolically apply distributive property (x + 1)(x + 3) = (x + 1)(x) + (x+ 1)(3) = x.x + 1.x + x.3 + 1.3 = x 2 + x + 3x + 3 = x2 + 4x + 3 x

  11. Using Graphing Calculator [-6, 6, 1] by [ -4, 4, 1 ] [-6, 6, 1] by [ -4, 4, 1 ]

  12. Some Special products Product of Sum and Difference (a + b) (a - b) = a 2 - b 2 Square of a Binomial (a + b) 2 = a 2 + 2ab + b 2 (a – b) 2 = a 2 - 2ab + b 2

  13. Squaring a binomial (Ex 11) (a + b) 2 = a2 + ab + ab + b2 = a2 + 2ab + b2 a a2 ab b ba b2 a b (a + b)2 = a2 + 2ab + b2

  14. 5.3Factoring Polynomials • Common Factors • Factoring by Grouping • Factoring and Equations ( Zero Product Property)

  15. Zero- Product Property For all real numbers a and b, if ab = 0, then a = 0 or b = 0 ( or both)

  16. Ex – 5 ( Pg 350 ) Solving the equation 4x –x2 = 0 graphically and symbolically Graphically Symbolically 4x – x 2 = 0 x(4 – x) = 0 (Factor out x ) x = 0 or 4 – x = 0 ( Zero Product property) x = 0 or x = 4 • Numerically • x y • -1 -5 • 0 0 • 3 • 4 • 3 • 0 • 5 -5 4 3 2 1 -3 -2 -1 1 2 3 4 5 6

  17. 5.4 Factoring TrinomialsFactoring x2 + bx + cx2 + bx + c , find integers m and n that satisfy m.n = c and m + n = bx2 + bx + c = (x + m)(x + n)

  18. Factor by Grouping Step 1 : Use parentheses to group the terms into binomials with common factors. Begin by writing the expression with a plus sign between the binomials. Step 2: Factor out the common factor in each binomial. Step 3: Factor out the common binomial. If there is no common binomial, try a different grouping

  19. 5.4 Factoring Trinomials ax 2 + bx + c by grouping Factoring Trinomials with Foil 3x 2+ 7x + 2 = ( 3x + 1 ) ( x + 2 ) x + 6x 7x If interchange 1 and 2 (3x + 2)(x + 1) = 3x2 + 5x + 2 which is incorrect 2x + 3x = 5x

  20. 5.5 Special types of Factoring Difference of Two Squares (a –b) (a + b) = a2- b2 (a + b) 2 = a2 + 2ab + b2 (a –b) 2 = a2 - 2ab + b2 Sum and Difference of Two Cubes (a + b)(a2 – ab + b2) = a3 + b3 (a –b)(a2 + ab + b2 ) = a3 – b3 Verify ( a + b) (a2 – ab + b2) = a. a2 – a . ab + a. b2 + b. a2 -b . ab + b. b2 = a3 - a2 b + a b2 + a2 b - a b2 + b3 = a3 + b3 Perfect Square

  21. 5.6 Factoring Polynomials Step 1: Factor out the greatest common factor, if possible. Step 2 : A. If the polynomial has four terms, try factoring by grouping. B. If the polynomial is a binomial , try one of the following. 1. a2- b2 = (a –b) (a + b) 2. a3 + b3 = (a + b)(a2 – ab + b2) 3. a3 – b3 = (a –b)(a2 + ab + b2 ) Otherwise , apply FOIL or grouping 4. (a + b) 2 = a2 + 2ab + b2 5. (a –b) 2 = a2 - 2ab + b2 Step 3: Check to make sure that the polynomial is completely factored

  22. 5.7 Polynomial Equations Solving Quadratic Equations Higher Degree Equations

  23. Using Graphing Calculator [ -4.7, 4.7, 1] by [-100, 100, 25] Y = 16x^4 – 64x^3 + 64x^2 x = 0 , y = 0, x = 2 y = 0

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