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Chapter 6

Chapter 6. Polynomials and Polynomial Functions. 6.1 Using Properties of Exponents. Properties of Exponents. Let a and b be real numbers and let m and n be integers. PRODUCT OF POWERS PROPERTY a m • a n = a m+n POWER OF A POWER PROPERTY ( a m ) n = a mn

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Chapter 6

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  1. Chapter 6 Polynomials and Polynomial Functions

  2. 6.1 Using Properties of Exponents

  3. Properties of Exponents Let a and b be real numbers and let m and n be integers. PRODUCT OF POWERS PROPERTY am• an= am+n POWER OF A POWER PROPERTY (am)n= amn POWER OF A PRODUCT PROPERTY (ab)m= ambm NEGATIVE EXPONENT PROPERTY ZERO EXPONENT PROPERTY a0 = 1, a ≠ 0 QUOTIENT OF POWERS PROPERTY POWER OF A QUOTIENT PROPERTY

  4. A number is expressed in Scientific Notation if it is in the form c X 10n where 1 ≤ c < 10 and n is an integer. Decimal to scientific Notation: 5,284,000 = 5.284 X 106 0.0000075 = 7.5 X 10-6 Scientific Notation to decimal: 3.45670123 X 108 = 345,670,123 6.546 X 10-5 = 0.00006546

  5. Classwork: Pg 326 – 328 (1-15,48,52,53) Homework: Pg 326 – 328 (16-46 every 3rd, 47,49-51,54-56)

  6. 6.2 Evaluating and Graphing Polynomial Functions

  7. A Polynomial Function : Leading Coefficient: Constant Term: a0 Degree: n Standard Form: terms are written in descending order of exponents from left to right

  8. A Polynomial Function in Standard Form: Leading Coefficient: Constant Term: 5 Degree: 3 Standard Form: terms are written in descending order of exponents from left to right

  9. Polynomial Classifications

  10. X 3 Synthetic Substitution Use synthetic Substitution to evaluate f(x)=2x4 - 8x2 + 5x – 7 when x = 3. Write the value of x and the coefficients of ƒ(x) as shown. Polynomial in standard form 2x4 + 0x3 - 8x2 + 5x – 7 x-value Coefficients 3 2 0 -8 5 -7 add 6 18 30 105 f(3) = 98 The value of f(x) is the last number written in the bottom right hand corner 2 6 10 35 98

  11. END BEHAVIOR FOR POLYNOMIAL FUNCTIONS For an>0 and n is even, both ends point up For an>0 and n is odd, left end points down and right end points up For an<0 and n is even, both ends point down For an<0 and n is odd, left end points up and right end points down See chart on pg 331

  12. Graphing Polynomial Functions To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. ƒ(x) = x3 + x2 - 4x – 1

  13. Classwork: Pg 333 – 336 (1-13, 65,68) Homework: Pg 333 – 336 (15,16,20,21,22,25,27,28,36,37,41,45,47,49-52,53,56,59,62,66,70, 80, 81)

  14. 6.3 Adding, Subtracting, and Multiplying Polynomials

  15. To add or subtract polynomials, add or subtract the coefficients of like terms. You can use a vertical or horizontal format. 3x3+ 2x2 – x - 7 + x3 - 10x2 - x + 8 (9x3 - 2x + 1) + (5x2 + 12x - 4) = 9x3 + 5x2 - 2x + 12x + 1 – 4 = 9x3 + 5x2 + 10x - 3 4x3 - 8x2 - x + 1 8x3 - 3x2 - 2x + 9 -2x3 - 6x2 + x - 1 6x3 – 9x2 – x + 8 8x3 - 3x2 - 2x + 9 -(2x3 + 6x2 -x + 1) Add the opposite. (2x2 + 3x) -(3x2 + x - 4)= 2x2 + 3x - 3x2 - x + 4 = - x2 + 2x + 4 Add the opposite.

  16. Multiplying Polynomials To Multiply polynomials, use the distributive property regardless if you use the vertical or horizontal method. Horizontally (x - 3)(3x2 - 2x - 4)=x(3x2)+x(-2x)+x(-4)-3(3x2)-3(-2x)-3(-4) =3x3 – 2x2- 4x – 9x2 + 6x + 12 =3x3 – 11x2 + 2x +12 • Vertically • - x2 + 2x + 4 • X x - 3 • 3x2 - 6x - 12 • x3 + 2x2 +4x • x + 5x2 – 2x - 12 Multiply -x2 + 2x + 4 by -3. Multiply -x2 + 2x + 4 by x. Combine like terms.

  17. Special Product Patterns SUM AND DIFFERENCE Example (a + b)(a - b) = a2 - b2 (x + 3)(x - 3) = x2 - 9 SQUARE OF A BINOMIAL (a + b)2 = a2 + 2ab + b2 (y + 4)2 = y2 + 8y + 16 (a - b)2 = a2 - 2ab + b2 (3t2 - 2)2 = 9t4 - 12t2 + 4 CUBE OF A BINOMIAL (a + b)3 = a3 + 3a2b + 3ab2 + b3 (x + 1)3 = x3 + 3x2 + 3x + 1 (a - b)3 = a3 - 3a2b + 3ab2 - b3 (p - 2)3 = p3 - 6p2 + 12p - 8

  18. Classwork: Pg 341 – 343 (1-12) Homework: Pg 341 – 343 (13 – 60 every 3rd, 62,63)

  19. 6.4 Factoring and Solving Polynomial Equations

  20. In Chapter 5 you learned how to factor the following types of quadratic expressions. TYPE EXAMPLE General trinomial 2x2 - 5x - 12 = (2x + 3)(x - 4) Perfect square trinomial x2 + 10x + 25 = (x + 5)2 Difference of two squares 4x2 - 9 = (2x + 3)(2x - 3) Common monomial factor 6x2 + 15x = 3x(2x + 5) In this lesson you will learn how to factor other types of polynomials.

  21. SPECIAL FACTORING PATTERNS SUM OF TWO CUBES Example a3 + b3 = (a + b)(a2 - ab + b2) x3 + 8 = (x + 2)(x2 - 2x + 4) DIFFERENCE OF TWO CUBES a3 - b3 = (a - b)(a2 + ab + b2) 8x3 - 1 = (2x - 1)(4x2 + 2x + 1) x3 + 27 = x3 + 33Sum of two cubes = (x + 3)(x2 - 3x + 9) 16u5 - 250u2 = 2u2(8u3 - 125) Factor common monomial. = 2u2[(2u)3 – 53] Difference of two cubes = 2u2(2u - 5)(4u2 + 10u + 25)

  22. Factor By Grouping The pattern for this is as follows. ra + rb + sa + sb = r(a + b) + s(a + b) = (r + s)(a + b) x3 - 2x2 - 9x + 18 = x2(x - 2) - 9(x - 2) Factor by grouping. = (x2 - 9)(x - 2) = (x + 3)(x - 3)(x - 2) Difference of squares

  23. In Chapter 5 you learned how to use the zero product property to solve factorable quadratic equations. You can extend this technique to solve some higher-degree polynomial equations. Solve 2x5 + 24x = 14x3. 2x5 + 24x = 14x3Write original equation. 2x5 - 14x3 + 24x = 0 Rewrite in standard form. 2x(x4 - 7x2 + 12) = 0 Factor common monomial. 2x(x2 - 3)(x2 - 4) = 0 Factor trinomial. 2x(x2 - 3)(x + 2)(x - 2) = 0 Factor difference of squares. x = 0, x = , x = - , x = -2, or x = 2 Zero product property The solutions are 0, , , -2, and 2. Check these in the original equation.

  24. Classwork: Pg 348 – 350 (1-3,5-16,88) Homework: Pg 348 – 350 (18,21,24,27-32,33,37,41,45,49,50,51,68,71,77,80,89)

  25. 6.5 The Remainder and Factor Theorems If a polynomial ƒ(x) is divided by x -k, then the remainder is r = ƒ(k). REMAINDER THEOREM A polynomial ƒ(x) has a factor x - k if and only if ƒ(k) = 0. FACTOR THEOREM

  26. Polynomial Long Division Divide 2x4 + 3x3 + 5x - 1 by x2 - 2x + 2. Subtract 2x2(x2 - 2x+2) Subtract 7x(x2 - 2x+2) Subtract 10(x2 - 2x+2) Remainder Answer

  27. Synthetic Division Divide x3 + 2x2 - 6x - 9 by x - 2 Solve for x: x-2=0 X=2 2 1 2 -6 -9 2 8 4 1 4 2 -5 When writing solution, reduce the degree of the polynomial by 1. Add the remainder divided by the divisor at the end of the answer Answer

  28. If the remainder is 0 then the divisor is a solution or a zero or a factor. You can use this fact to help factor completely.

  29. Classwork: Pg 356-358 (4-13,39,47) Homework: P 356 – 358 (15-54 every 3rd)

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