Ch+6+0910

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Answers to practice problems
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**Text (Notes) - Amanda Saravia and Gabby Sallada** Real-Life Applications- Devin Taylor Multimedia - Alex schmitt
 * Practice Problems - Cheyenne Stambaugh and Mary Sweitzer

SECTION 6.1 Properties of Exponents **


 * ** Name Of Property ** || ** Algebra ** || ** Example ** ||
 * *Product Of Powers* || a^m x a^n = a^m+n || 2^3 x 2^4 = 27 ||
 * Power Of A Power || (a^m)^n = a^mn || (2^4)^2 = 2^8 ||
 * Power Of A Product || (ab)^m = a^m x b^m || ( 3 x 4)^2 = 3^2 x 4^2 ||
 * Negative Exponent || a^-m= 1/a^m || 4^-2 = 1/4^2 ||
 * Zero Exponent || a^0 = 1, a ≠ 0 || 4^0 = 1 ||
 * *Quotient Of Powers* || a^m/a^n = a^m-n || 5^2/5^1 = 5^1 ||
 * Power Of A Quotient || (a/b)^m = (a^m/b^m) || (6/9)^2 = (6^2/9^2) ||
 * ** *Stars mean that bases need to be the same.* ** ||

A number expressed in the form c x 10^n where c is greater than or equal to one and less than ten and n is an integer. Example: .4500 4.5 x 10^3​
 * __ Scientific Notation __**

Evaluate the following expressions. Name the property that is used. 1. (2^3)^4 2. (3/4)^2 3. (-5)^-6(-5)^4 4. x^0 5. (7b^-3)^2b^5b
 * __PRACTICE PROBLEMS __**

Word Problem: An adult human body contains about 75,000,000,000,000 cells. Each cell is about 0.001 inch wide. If the cells were laid end to end to form a long chain, about how long would the chain be in miles? Give the answer in scientific notation. Real-Life Applications- Properties of Exponents can be used to find the ratio of a state's park space to total area Polynomials-a sum of terms where each term is a constant times a whole number power of x. __Standard Form -__ when in standard form equation will have highest exponent of variable first and others will be in decreasing order.
 * Section 6.2 **

Polynomials are classified by degrees. Constant -- 0 -- 4x^0=4(1)=4 Linear -- 1 --- 1/2x+3 Quadratic - 2 --- 4x^2+3x+2 Cubic -- 3 -- 5x^3+4x^2+x+3 Quartic 4 --- 8x^4+2x^3+x^2+x+3 Quintic 5 --- x^5-4
 * __ NameDegree-Example __**

Leading coefficent is the number in front of a value with the greatest exponent ex. -3x^4+1/2x^2-7 leading coefficent is -3

*Constant*=highest exoponent
 * exponents **__// must be //__** positive integers*

//*Polynomial functions// ex. f(x)= 5x+7 ex. f(x)= x^2+4x-7 ex. 3/2x^4-2/3x^2 //*Not Polynomial functions// ex. f(x)= 2x+3/5x-2 ex. 2x^-3 ex. f(x)= 1/x^2-5x+3

-** Monomial- ** 1 term -** Binomial -** 2 terms
 * -Trinomial -** 3 terms

__ Steps __ 1.) substitute x value in for x in the equation 2.) solve equation ex.4x^4+3x^2+2x-1, x=4 4(4)^4=3(4)^2+2(4)-1 4(256)+3(16)+8-1 1024+48+7 1079
 * Direct Substitution **-when sustituting the x value in for x in order to sove the equation

Synthetic Substitution -will be using coefficents and a given value for x __ Steps __ 1.) write the value of x and the coefficents from the equation in decreasing order from left to right 2.) bring down the leading coefficent and multiply it by the x value 3.) write the answer in the next column and add the numbers in that column 4.) write the sum below the line and multiply it by the x value and put the product in the next colmn 5.) continue to add and multiply until the last coefficent has been added to a number and that gives you your answer ex. 4x^4+3x^2+2x-1 __ 4--0--3--2--(-1) __ __-(16)---(64)--(268)--(1080)__ (4)--(16)--(67)--(270)-(1079) final answer: 1079 __2-0(-8)-5-(-7) -(6)---(18)---(30)--(105)__ 2---(6)---(10)--(35)(98) final answer: 98

// End Behavior of Polynomial Functions- // end behavior of a polynomial function's graph is the behavior of a graph as x approches positive infinity(+oo) or negative infinity(-oo) f(x) is y up to +oo(infinity) or down to -oo x is positive +oo(right) or negative -oo(left) ex. **Put Graph here** ex. **Put different graph, unlike first one, **

= Graphing Polynomials = 1.) First, use a table of values to get started-> x I y 2.) Then,plug the different x values from the -3 I ? table of values into the equation and put -2 I ? the different answers into the table ofvalues. -1 I ? 3.) Lastly, plot the points from thetable of values 0 I ? onto the graph 1 I ?

2 I ?

3 I ? .


 * ex. PUT GRAPH HERE FOR EQUATIONS F(X)=4X+2 AND F(X)= -X^4-2X^3+2X^2+4X**



**__Practice Problems for 6.2__** State whether or not the given function is a polynomial. If so, then state the degree, type, and leading coefficient. 1. f(x)= 12-5x 2. f(x)= x+ π 3. f(x)= -2x^3-3x^-3+x 4. f(x)= x^2-x+1 5. f(x)= 3x^3

When x=5, evaluate the given polynomial functions using direct substitution. 1. f(x)= 2x^3+5x^2+4x+8 2. f(x)=x+1/2x^3 3. f(x)=5x^4-8x^3+7x^2 4. f(x)=11x^3-6x^2+2 5. f(x)=7x^3+9x^2+3x

When x=-3, evaluate the following fuctions using synthetic division. 1.f(x)= 3x^5+5x^3+2x+9 2. f(x)=x^2+1/2x-3 3. f(x)=x^4-6x^2+7x 4. f(x)=4x^3+x^2+2 5. f(x)=7x^4+5x^3+3x

Graph each function, then describe its end behavior. 1.-5x^3-2 2.x^2+1 3.x^5-2x^2+2

Real-Life Application- Evaluating and Graphing can be used in real life for example by finding out how much money is awarded at the U.S. open. **Combining Polynomials // Adding //** Add __like terms__-varibles and exponents are the same->ex. (3x^2+7+x)+(14x^3+2+x^2-X)=14x^3+4x^2+9-->__Adding horizontially__ ---standard form-> 3x^2+x+2 adding vertically ->__+14x^3+x^2-x+2__ answer-->14x^3+4x^2+9

**Subtracting ** 8x^3-3x^2-2x+9 8x^3-3x^2-2x+9 //6x^3-9x^2-x+8//
 * -2x^3-6x^2+x-1**
 * __-(2x^3+6x^2-x+1)__<-**add the opposite

(2x^2+3x)-**(3x^2+x-4)**=2x^2+3x**-3x^2-x+4**=//-x^2+2x+4//

**// Multiplication //** When you multiply, all you do is foil the problem out and then combine like terms and put in standard form. ex. (4x - 5) (2x^5 + x^3 -1) FOIL->8x^6 + 4x^4 - 4x - 10x^5 - 5x^3 + 5 8x^6 - 10x^5 + 4x^4 - 5x^3 - 4x +5

__*Box Method*__ (3x-4)(8x-1) __l 3x l -4 8x l 24x^2 l -32x__ -1 l -3x l 4 then, add like terms and you get 24x^2-35x+4 //*when multiplying any kind of 2 polynomials box method can be used*// **// Long Division //** First put the problem into standard form. Then divide like you would divide a regular problem. If you are left with a remainder or not zero put that over the divisor. __2x_+_1_+_5 /x+4__ x + 4 ) 2x^2 + 9x + 9 __- 2x^2 - 8x__ x +9 __-x - 4__ 5


 * // Synthetic Division //**

The number in the box should make the linear binomial equal zero. It is an intercept. The boxed number is a remainder. To do synthetic division you do the same as synthetic substitution.

ex. -3] 6 -5 -6 __(-18 69__ -(6 -23 [63]

6x - 23 + 63 /x + 3

Real-Life Applications- Combining polynomials can be used in real life by seeing how much power is needed to keep a bike moving at a certain speed. Remainder Theorem - If a polynomial f(x) is divided by ( x - k ), then the remainder is f(k) = r. In other words, when you divide using synthetic division, such as below, the remainder will bethe same as when you use direct substitution.
 * // Theorems //**
 * ex. ( 3x^2 + 5x - 2) / ( x + 4)

__4]__ 3 5 -2 __-12 28__ 3 -7 26

3(-4^2) + 5(-4) - 2 3(16) - 20 - 2 48 - 2 - 2 26 **

Factor Theorem- A polynomial f(x) has a factor ( x - k ) if and only if f(k) = 0. **ex. x^2-4 (x+2)(x-2)**

__ **-2] 1 0 -4** __ **(-2 4 -(1 -2** **__ [0

2] 1 0 -4__ ---(2 4 (1 2** __ ** [0 ** __

-2 and 2 are the "zeros" of the polynomial. they also are the x intercepts

ex. 2x^3+5x^2-x+7 when x=2

2(2)^3+5(2)^2-2+7 32+20+5= 41

__2] 2 5 -1 7__ ---(4 18 34 ---(2 9 17 __[41__

Rational Zero Theorem - p/q = factor of constant term a0 / factor of leading coefficient

To find the zero use the p/q test to find all possible zeros. Then use synthetic division to see if one of the possible zeros is a zero. Next the equation you get from synthetic division needs to be factored. Once its factored you can solve for the rest of the zeros.

ex. f(x) = x^3 + 2x^2 - 11x - 12 p/q = +/- 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) zeros are -1, 3, and -4

Fundamental Theorem - 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 numbers.

Imaginary roots always come in pairs.

Real-Life Applications- In factoring and solving is finding the dimensions of a block discovered underwater.

Let f(x) = anx^n + an - 1x^n-1 + ... + a1x + a0. The following statements are equivalent. Zero - k is zero of the polynomial function f Factor - x - k is a factor of the polynomial f(x) Solution - k is a solution of the polynomial equation f(x) = 0. If k is a real number, then the following is a also equivalent. X intercept- k is an x-intercept of the graph of the polynomial function f. Local Maximum - the y coordinate of a turning point that is higher than all nearby points. Local Minimum - the y coordinate of a turning point that is lower than all nearby points.
 * Section 6.8 **

**Turning Points of Polynomial Functions** The graph of every polynomial function of degree n has at most n - 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly n - 1 turning points.

Practice Problems for 6.3: a.(-3x^3+x-11) - (4x^3+x^2-x) b.(10x^3-4x^2+3x) - (x^3-x^2+1) c.(10x-3+7x^2) + (x^3-2x+17)
 * 1) 1.Find the sum or difference.

a.x(x^2+6x-7) b.-4x(x^2-8x+3) c.(x-4)(x-7)
 * 1) 2. Multiply the polynomials.

a.(x+9)(x-2)(x-7) b.(x+5)(x+7)(-x+1) c.(x-9)(x-2)(3x+2)** 
 * 1) 3. Multiply the three binomials.

Practice Problems for 6.4: Factor the polynomial using any method. 1.x^6+125 2.X^4-1 3.5x^3-320 4.3x^2+11x+6 5.125x^3-216

Find the GCF 1.3x^4-12x^3 2.24x^4-x^6 3.145x^9-17

Factor. 1. x^3-8 2. 216x^3+1 3. 1000x^3+27

Factor by Grouping 1. x^3+x^2+x+1 2.x^3+3x^2+10x+30 3.2^3-5x^2+18x-45

Find the real number solutions. 1.2x^3-6x^2=0 2.x^3+27=0 3.x^4+7X^3-8x-56=0 4.3X^7-243x^3=0 5.8x^3-1=0  Real-Life Applications- In Factoring and Solving polynomial equations is finding the dimensions of a block discovered underwater

Practice Problems for 6.5: a. Dividex^3+2x^2-6x-9 by x+3 b. Divide2x^4+3x^3+5x-1 by x-2 c. Divide 4x^3-x^2+5x-1 by x-4
 * 1) 1. Use Synthetic Division.

a.f(x)=2x^3+11x^2+18x+9 k= -3 b.f(x)=x^3-5x^2-2x+24 k= -2 c.f(x)=4x^3-4x^2-9x+9 k=1
 * 1) 2.Find the factors of the polynomial given that f(k)=0.

a. One zero of f(x)=x^3-2x^2-9x+18 is x=2 b. One zero of f(x)=9x^3+10x^2-17x-2 is x= -2 c. One zero of f(x)=2x^3+3x^2-39x-20 is x=4
 * 1) 3.Find the other zeros of the polynomial function.

a.(x^2+7x-5) / (x-2) b.(2x^2+3x-1) / (x+4) c.(x^2+5x-3) / (x-10) Real-Life Applications- In in the Remainder an Factor Theorems is used in finding a production level that yields a certain profit
 * 1) 4. Use polynomial long divison.

Practice Problems for 6.6: a.f(x)=x^3+2x^2-11x-12 b.f(x)=x^4+2x^2-24 c.f(x)=2x^5+x^2+16
 * 1) 1.Find all possible rational zeros.

a.f(x)=x^3+7x^2-4x-28 b.f(x)=x^4+3x^3-7x^2-27x-18 c.f(x)=x^4+3x^3+3x^2-3x-4
 * 1) 2. Using sythetic divison decide which of the following are zeros of the function; 1, -1, 2, -2.

a.f(x)=x^3-8x^2-23x+30 b.f(x)=x^3-7x^2+2x+40 c.f(x)=x^3+72-5x^2-18x Real-Life Applications- In Finding Rational Zeros can be used to find the dimensions of a monument 6.7 Real-Life Applications- In Fundamental Theorem is used in real life by finding the population of certain people Practice Problems for 6.8: do numbers 23, 25 and 27.
 * 1) 3.Find all the real zeros of the function.
 * 1) 1. Estimate the coordinates of each turning point and state whether each cooresponds to a local maximum or a local minimum. Then list all the real zeros and determine the least degree that the function can have.

a.f(x)=3x^3-9x+1 b.f(x)= -1/4x^4+2x^2 c.f(x)=x^5-5x^3+4x
 * 1) 2.Use a graphing calculator to graph the polynomial function. Identify the x-intercepts and the points where te local maximums and local minimums occur.

a.f(x)=(x-1)^3(x+1) b.f(x)=1/8(x+4)(x+2)(x-3) c.f(x)=5(x-1)(x-2)(x-3)
 * 1) 3. Graph the function.

Real-Life Applications- In Analyzing graphs is used to find the certain crop productions in the United States media type="file" key="jan14.mov" width="300" height="300"

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