Chapter+6

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= Chapter 6- Polynomials and Polynomial Functions = = = 

=6.1-Using Properties of Exponents =


 Properties of exponents- PRODUCT OF POWERS PROPERTY : a^m * a^n = a^m+n PO WER OF A POWER PROPERTY: (a^m) = a^mn POWER OF A PRODUCT PROPERTY: (ab)^m = a^m * b^m NEGATIVE EXPONENT PROPERTY: a^-m = 1/a^m, "a" cannot be 0 QUOTIENT OF POWERS PROPERTY: a^m/a^n = a^m-n, "a" cannot be 0 POWER OF A QUOTIENT PROPERTY: (a/b)^m = a^m/b^m, "b" cannot be 0

Scientific Notation- <span style="color: rgb(0, 21, 255);">SCIENTIFIC NOTATION: when a number is written in the form c * 10^n where c is greater than or equal to 1 and is less than 10 and n is an integer For example, the width of a molecule of water is about 2.5 X 10^-8 meter, or 0.000000025 meter. When working with numbers in scientific notation, the properties of exponents can help with calculations.

=<span style="color: rgb(237, 84, 84);">6.2-Evaluation and Graphing Polynomial Functions =

<span style="color: rgb(0, 21, 255);">POLYNOMIAL FUNCTION : a function in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e where a cannot be 0, the exponents are all whole numbers, and the coefficients are all real numbers. In this polynomial function a is the <span style="color: rgb(0, 21, 255);">LEADING COEFFICIENT, e is the <span style="color: rgb(0, 21, 255);">CONSTANT TERM , and 4 is the <span style="color: rgb(0, 21, 255);">DEGREE. A polynomial in <span style="color: rgb(0, 21, 255);">STANDARD FORM is written so as all of its terms are in descending order of exponents from left to right.

<span style="color: rgb(237, 69, 231);">Examples of Degree - f(x) = 3x + 2 is a polynomial function of degree 1 f(x) = x^2 + 3x +2 is a polynomial function of degree 2

<span style="color: rgb(237, 69, 231);">Common Types of Polynomial Functions- media type="custom" key="2967406" http://www.teachertube.com/view_video.php?viewkey=e258cee61f4377e5bf3e
 * Degree || Type || Standard Form Example ||
 * 0 || Constant || f(x) = a ||
 * 1 || Linear || f(x) = 3x + 4 ||
 * 2 || Quadratic || f(x) = 5x^2 + 3x + 2 ||
 * 3 || Cubic || f(x) = 11x^3 + 1/2x^2 + 5x + 9 ||
 * 4 || Quartic || f(x) = 9x^4 + 7x^3 + 22x^2 + 90x + 1 ||

<span style="color: rgb(237, 69, 231);">Methods for Solving- <span style="color: rgb(0, 21, 255);">DIRECT SUBSTITUTION - substitute the given value of x into the polynomial function in order to solve for f(x) <span style="color: rgb(0, 21, 255);">SYNTHETIC SUBSTITUTION - write the given value of x and take note of the coefficients of f(x). Bring down the leading coefficient and multiply it by the given x value then add the result to the next coefficient in order. Take that sum and multiply that by the given x value and add it to the next coefficient, continue this until you are left with the solution.

<span style="color: rgb(237, 69, 231);">Example of Synthetic Substitution- f(x) = 2X^4 - 8X^2 + 5X - 7, when x = 3 Drop down leading coefficients: 2 0 -8 5 -7 Multiply them by x(3) and add: 6 18 30 105 Added totals (solution is last) :2 6 10 35 98 Solution when x = 3: f(3) = 98

<span style="color: rgb(237, 69, 231);">Graphing Polynomial Functions- <span style="color: rgb(0, 21, 255);">END BEHAVIOR- the behavior of the graph as x approaches positive infinity ( + ∞) or negative infinity ( -∞). The expression x--> +∞ is read as "x approaches positive infinity."

<span style="color: rgb(237, 69, 231);">End Behavior for Polynomial Functions (based on the degree and leading coefficient)- Negative leading coefficient and even degree: x--> +∞, f(x)--> -∞ and x--> -∞, f(x)--> -∞ Positive leading coefficient and even degree: x--> +∞, f(x)--> +∞ and x--> -∞, f(x)--> +∞ Negative leading coefficient and odd degree: x--> +∞, f(x)--> -∞ and x--> -∞, f(x)--> +∞ Positive leading coefficient and odd degree: x--> +∞, f(x)--> +∞ and x--> -∞, f(x)--> -∞

<span style="color: rgb(237, 69, 231);">To graph: 1. Determine the end behavior of the graph to help check the graph. 2. Pick a few points around the origin to plug into the function and solve for. 3. Graph those points and see if they match your end behavior, if so, good job! media type="custom" key="2967418" http://www.metacafe.com/watch/2110596/college_algebra_graphing_polynomial_functions/

= <span style="color: rgb(237, 84, 84);">6.3- Adding, Subtracting, and Multiplying Polynomials <span style="color: rgb(237, 84, 84);"> = =<span style="color: rgb(237, 84, 84);"> = <span style="color: rgb(237, 84, 84);">

<span style="color: rgb(237, 69, 231);">Adding <span style="color: rgb(0, 21, 255);">To add polynomials, all you do is combine like terms, it is always helpful to align like terms in a verticle column as well. For example: 5x^4 - 2x^3 + 10x^2 - 7x - 17 + 6x^3 - 14x^2 + 20 =5x^4 + 4x^3 - 4x^2 - 7x + 3

<span style="color: rgb(237, 69, 231);">Subtract <span style="color: rgb(0, 21, 255);">To subtract polynomials, you change the subtraction sign in the problem to an addition sign and make the terms following the new addition sign opposite of what they were. For example: (5x^2 + 4x^3 - 7) - (4x - 7x^5 + 9x^2) original problem (5x^2 + 4x^3 - 7) + (-4x + 7x^5 -9x^2) switch signs 7x^5 + 4x^3 - 4x^2 - 4x - 7 add like terms

<span style="color: rgb(237, 69, 231);">Multiply <span style="color: rgb(0, 21, 255);">To multiply polynomials use foil to multiply everything together. For example: (x - 3) (4x^2 - 5) (2x^3 +1) original problem 4x^3 -5x -12x^2 +15 (2x^3 +1) foil any pair of terms 8x^6 + 4x^3 - 10x^4 - 5x - 24x^5 - 12x + 30x^3 +15 foil the terms left 8x^6 - 24x^5 - 10x^4 + 34x^3 -17x + 15 combine like terms

<span style="color: rgb(237, 69, 231);"> Special Product Patterns <span style="color: rgb(0, 21, 255);">Example <span style="color: rgb(0, 21, 255);">Sum and Difference- (a + b)^2 (a - b)^2 = a^2 - b^2 (x - 3)(x + 3) = x^2 - 9 <span style="color: rgb(0, 21, 255);">Square of a Binomal- (a + b)^2 = a^2 + 2ab + b^2 (y + 4)^2 = y^2 + 8y + 16 (a - b)^2 = a^2 - 2ab + b^2 (3t^2 - 2)^2 = 9t^4 - 12 t^2 + 4 <span style="color: rgb(0, 21, 255);">Cube of a Binomial- (a + b)^3 = a^3 + 3a^2b + 3ab^2 =b^3 (x + 1)^3 = x^3 + 3x^2 + 3x + 1 (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 (p - 2)^3 = p^3 - 6p^2 + 12p - 8

=<span style="color: rgb(237, 84, 84);">6.4- Factoring and Solving Polynomial Equations =

<span style="color: rgb(237, 84, 231);">Factoring (*** are the new ones):**
 * **Type** || **Example** ||
 * **General Trinomial** || 2x^2 - 5x - 12 = (2x + 3)(x - 4) ||
 * **Perfect Square Trinomial** || x^2 + 10x + 25 = (x + 5)^2 ||
 * **Difference of Two Squares** || 4x^2 - 9 = (2x + 3)(2x - 3) ||
 * **Common Monomial Factor** || 6x^2 + 15x = 3x(2x + 5) ||
 * ***Sum of Two Cubes** || a^3 + b^3 = (a + b)(a^2 - ab + b^2) ||
 * ***Difference of Two Cubes** || a^3 - b^3 = (a - b)(a^2 + ab + b^2) ||
 * <span style="color: rgb(0, 21, 255);">Example of Sum of Two Cubes- x^3 + 8 = (x+ 2)(x^2 - 2x + 4)
 * <span style="color: rgb(0, 21, 255);">Example of Difference of Two Cubes**-** 8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1)

<span style="color: rgb(237, 84, 231);">Solving Polynomials by Factoring- Solving polynomials in this section is just about identical to the way you are used to, once the equation is fully factored, set all terms equal to 0 and solve, this will give you the x solutions.

<span style="color: rgb(0, 21, 255);">Example- 3x^3 - 12x^2 + 2x - 8 = 0 3x^2(x - 4) + 2(x - 4) = 0 (3x^2 + 2)(x - 4) = 0 3x^2 + 2 = 0, x = i √2/3 and -i√2/3 x - 4 = 0, x = 4 x = 4, i √2/3, -i √2/3 media type="custom" key="3048722" <span style="color: rgb(237, 84, 84);">

=<span style="color: rgb(237, 84, 84);">6.5 The Remainder and Factor Theorems =

<span style="color: rgb(237, 69, 231);">The Remainder and Factor Theorems When you divide a polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) and the remainder polynimial r(x). We write this as f(x)/d(x) = q(x) + r(x)/d(x). The degree of the remainder must be less than the degree of the divisor. <span style="color: rgb(0, 21, 255);">Remainder Theorem- if a polynomial f(x) is divided by x - k, then the remainder is r = f(k).

<span style="color: rgb(237, 69, 231);">Polynomial Long Division- Divide 2x^4 + 3x^3 + 5x - 1 by x^2 - 2x + 2

<span style="color: rgb(130, 0, 255);"><span style="color: rgb(0, 0, 0);">--> 2x^2 + 7x + 10 -> <span style="color: rgb(136, 195, 19);">x^2 - 2x + 2 | <span style="color: rgb(217, 189, 58);">2x^4 + 3x^3 + 0x^2 + 5x - 1 >+ -2x^4 + 4x^3 - 2x^2 > = 7x^3 - 2x^2 + 5x - 1 --->+ -7x^3 + 14x^2 -14x + 0 --->= 10x^2 - 9x - 1 --->+ -10x^2 + 20x - 20 --->= <span style="color: rgb(223, 59, 247);">11x - 21

<span style="color: rgb(217, 189, 58);">2x^4 + 3x^3 + 5x - 1 / <span style="color: rgb(136, 195, 19);">x^2 - 2x + 2 =<span style="color: rgb(0, 0, 0);"> <span style="color: rgb(130, 0, 255);">2x^2 + 7x + 10 + <span style="color: rgb(223, 59, 247);">11x -21 /<span style="color: rgb(136, 195, 19);">x^2 - 2x + 2

Always use first terms, no other part of the polynomials(terms with the highest powers) Figure out what times the first term of the divisor willl get it closest to the first term of the divisor and write that up top. Then multiply all the terms in the divisor by that number and subtract that from the dividend. Continue to do this until you are left with a remainder polynomial with a degree that is less than the divisor. Write the remainder polynomial over the divisor The solution can be checked by multiplying the divisor by the quotient and adding the remainder, the result should be the dividend.

<span style="color: rgb(237, 69, 231);">Synthetic Division- Synthetic division is much like synthetic substitution. When you have a binomial divisor, you just find the zero of that expression and use that number to multiply the others by in your synthetic division. (The zero of the expression x - 2 is 2, because if x = 2, then the whole expression will equal 0). The bottom row of numbers that result from your synthetic division are the coefficients of the polynomial quotient adn the last number on the right is the remainder.

Example- x^4 - 2x^3 + 5x + 2 = 0 divided by x - 4, the zero would be 4

4] -->1 -2 0 5 2 ->4 8 32 148 -> 1 2 8 37 150 Therefore the quotient is: x^3 + 2x^2 + 8x + 37 + 150/x-4 media type="custom" key="3048764"
 * 150/x-4 is the remainder

=<span style="color: rgb(237, 84, 84);">6.6 Finding Rational Zeros =

<span style="color: rgb(0, 21, 255);">Rational Zero Theorem- If f(x) = ax^n + ...... bx + c has integer coefficients, then every rational xero of f has the following form: p/q = factors of constant term (c) / factors of leading coefficient (a^n)

Example: Fing the rational zeros of f(x) = x^3 + 2x^2 - 11x - 12

List the possible rational zeros, The leading coefficient is a 1 and the constant is -12. The possible rational zeros are: x = +/- (1, 2, 3, 4, 6, 12)

Test these possible zeros by synthetic division:

__1|__ 1 2 -11 -12 | __1 3 -8__  1 3 -8 -20  There is a remainder, therefore 1 is not a zero. __-1|__ 1 2 -11 -12 | __-1 -1 12__  1 1 -12 0  There is no remainder, therefore -1 is a zero. Since -1 is a zero of f, you can write the following: f(x) = (x + 1)(x^2 + x -12)

This can be factored: f(x) = (x + 1)(x - 3)(x + 4)

Therefore the zeros of f are -1, 3, and -4.


 * HINT**- The original polynomial equation can be plugged into the y= screen of a graphing calculator to find the zeros as well.

=<span style="color: rgb(237, 84, 84);">6.7 Using the Fundamental Theorem of Algebra =

<span style="color: rgb(0, 21, 255);">The Fundamental Theorem of Algebra- If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex mumbers. (A polynomial will have the same number of solutions as the degree of the polynomial = x^3 + 7 = f(x) has 3 solutions)

<span style="color: rgb(0, 21, 255);">Repeated Solution - a solution of a polynomial that repeats, but is still counted towards the total number of solutions.

Example- Find all of the zeros of: f(x) = x^5 - 2x^4 + 8x^2 - 13x + 6 f(x) = (x - 1)(x - 1)(x + 2)(x^2 - 2x + 3) f(x) = (x - 1)(x - 1)(x + 2)[x - (1 + i √2)][x - (1 - i√2)] x = 1, 1, -2, 1 + i √2, 1 - i√2 This polynomial has 5 solutions and is of the 5th degree.


 * Real zeros** will cross the x-axis, however imaginary solutions will not cross the x-axis.

media type="custom" key="3092980"

=<span style="color: rgb(237, 84, 84);">6.8 Analyzing Graphs of Polynomial Functions =

Summary of Concepts:

<span style="color: rgb(0, 21, 255);">Zero - a number k is a zero if f(x) = 0 <span style="color: rgb(0, 21, 255);">Factor- x - k is a factor of f(x) <span style="color: rgb(0, 21, 255);">Solution - k is a solution of the polynomial equation f(x) = 0. If k is a real number, then the following is also equivalent. <span style="color: rgb(0, 21, 255);">X-intercept - k is an x-intercept of the graph of the polynomial function f (k is a zero)

The key concepts learned in previous chapters can be used to graph polynomial functions.
 * Factoring can be used to find the x-intercepts of the polynomial's graph
 * After finding those, you can determine the end behavior of the polynomial to be checked with the graph later
 * Plot those intercepts and if needed use a table to find other points on the graph
 * Check the end behavior with your final graph

<span style="font-size: 140%; color: rgb(0, 22, 255);">Practice Problems <span style="font-size: 110%; color: rgb(233, 31, 22);"> **<span style="font-size: 120%; color: rgb(223, 12, 12);">6.1 Using properties of Exponents ** Example: (8^2)^3 Solve: (8^2)^3 = 8^2+3 = 8^5 = 8×8×8×8×8 = 32768 Now you try: 17. (5^-2)^3 21. (3/7)^3 Answers: The Back of Your Book p SA 20 **<span style="font-size: 110%; color: rgb(229, 11, 11);"><span style="font-size: 110%; color: rgb(240, 20, 20);">6.2 Evaluating and Graphing Polynomial <span style="color: rgb(225, 14, 14);">Functions  ** Example Direct Substitution - f(x)=2x^4 - 8x^2 + 5x - 7, where x=2 =2(2)^4 - 8(2)^2 + 5(2) - 7 =32 - 32 + 10 - 7 =3 Now you try: 33. 11x^3 - 6x^2 + 2 where x=0 35. 7x^3 + 9x^2 + 3x where x=10 Answers: The Back of Your Book, p SA 21 Adding Example - (3x^3 + 2x^2 - x - 7) + (x^3 - 10x^2 + 8) = 4x^3 + 8x^2 - x + 1 Now you try: 19. (4x^2 - 11x + 10) + (5x - 31) 25. (10x - 3 + 7x^2) + (x^3 - 2x +17) Answers: The Back of Your Book p. SA 21
 * <span style="font-size: 120%; color: rgb(215, 20, 20);">6.3 Adding, Subtracting and Multiplying Polynomials **

Subtracting Example - (8x^3 - 3x^2 - 2x +9) - (2x^3 + 6x^2 - x + 1) = (8x^3 - 3x^2 - 2x +9) + (-2x^3 - 6x^2 + x - 1) = (6x^3 - 9x^2 - x +8) Now you try: 23. (10x^3 - 4x^2 + 3x) - (x^3 - 2x + 17) 15. (x^2 - 6x + 5) - (x^2 + x - 2) Answers: The Back of Your Book p. SA 21

Multiplying Example: (x - 3)(3x^2 - 2x - 4) = 3x^3 - 11x^2 + 2x + 12 Now you try: 41. (3x^2 - 2)(x^2 + 4x + 3) 39. (x - 1)(x^3 + 2x^2 + 2) Answers: The Back of Your Book p.SA 21

<span style="font-size: 120%; color: rgb(219, 6, 6);"><span style="color: rgb(0, 0, 0);"><span style="font-size: 120%; color: rgb(233, 7, 7);">6.4 Factoring and Solving Polynomial Equations Example: 2x^5 + 24x = 14x^3 2x^5 + 24x - 14x^3 = 0 **Bold** - the bolded is the solution to this if it were a factoring problem 2x(x^4 - 7x^2 + 12) = 0 2x(x^2 - 3)(x^2 - 4) = 0 x = 0, ± square root of 3, -2, 2
 * 2x(x^2 - 3)(x - 2)(x + 2)** = 0

Now you try: 79. x^3 - 15x^2 + 5x - 25 = 0 61. 4x^4 + 39x^2 - 10 <span style="color: rgb(233, 7, 7);">**6.5 The Remainder and Factor Theorems** <span style="color: rgb(0, 0, 0);"> <span style="color: rgb(0, 0, 0);">   Example: (x^3 + 2x^2 - 6x - 9) / (x -2)

__2|__ 1 2 -6 -9 2 8 4 ¯¯¯¯¯¯¯¯¯¯¯ 1 4 2 -5 x^2 + 4x + 2 + (-5/x-2)

Now you try: 31. (2x^2 + 7x + 8) / (x - 2) 35. (10x^4 + 5x^3 + 4x^2 - 9) / (x + 1) Answers: The Back of Your Book p SA 22     <span style="color: rgb(233, 7, 7);">

Example: f(x) = x^3 + 2x^2 - 11x - 12 all possible zeros: ±1, ±2, ±3, ±4, ±6, ±12 __-1|__ 1 2 -11 -12 __-1 -1 12__ 1 1 -12 0 f(x) = (x + 1)(x^2 + x - 12) f(x) = (x + 1)(x - 3)(x + 4) x = -1, 3, -4
 * 6.6 Finding Rational Zeros  **

Now you try: 47. f(x) = 2x^3 + 4x^2 - 2x - 4 53. f(x) = 2x^4 + 3x^3 - 3x^2 + 3x - 5 Answers: The Back of Your Book p SA 22 <span style="color: rgb(233, 7, 7);">**6.7 Using the Fundamental Theorem of Algebra** <span style="color: rgb(233, 7, 7);"> <span style="font-size: 110%; color: rgb(3, 2, 2);">Example: f(x) = x^5 - 2^4 + 8x^2 - 13x + 6 all possible zeros: ±1, ±2, ±3, ±6 <span style="font-size: 110%; color: rgb(3, 2, 2);"> ***after synthetic division (to show the theorem rather than repeat 6.6)*** f(x) = (x - 1)(x - 1)(x + 2)(x^2 - 2x + 3) f(x) = (x - 1)(x - 1)(x + 2)[x - (1 + i|¯2)][x - (1 - i |¯2)] x = 1, 1, -2, 1 ± i|¯2

Now you try: 29. x^4 + 6x^3 +14x^2 +54x + 45 31. x^4 - x^3 - 5x^2 - x - 6 Answers: The Back of Your Book p SA 23 <span style="font-size: 120%; color: rgb(241, 9, 9);">6.8 Analyzing Graphs of Polynomial Functions Example: __x | y__ -4 | -12 (3/4) -3 | -4 -1 | 1 0 | (1/2) 2 | 1 3 | 5
 * insert graph here*** (due to technical difficulty, the graph before you is nonexistent)

Now you try: graph the functions 17. f(x) = 5(x - 1)(x - 2)(x - 3) 21 f(x) = (x - 2)(x^2 + x + 1) Answers: The Back of Your Book p SA 23

<span style="color: rgb(229, 11, 11);"> Careers That Use Polynomials Wildlife biologists use a lot of polynomials <span class="Apple-style-span" style="color: rgb(128, 0, 128);"> with their job, such as Ornithologists, who study birds. They collect large amounts of data and information that they use to describe bird populations and characteristics, and they create mathematical models that help determine the size and growth of bird populations. People <span class="Apple-style-span" style="color: rgb(128, 0, 128);"> in the medical field, like nurses, also use math on their job when dealing with medication,taking vital signs, making assessments, and <span class="Apple-style-span" style="color: rgb(128, 0, 128);"> helping to make treatment plans. Gerontologists <span class="Apple-style-span" style="color: rgb(0, 0, 128);">use math in their jobs when providing financial advice, creating budgets, or when studying demographic data and interpreting statistics. <span class="Apple-style-span" style="color: rgb(128, 0, 128);"> Archeologists develop and test hypotheses based on material remains, and statistics are used when they compile and analyze large amounts of data gathered during their fieldwork. There are many careers that use polynomials with their job.