Project in Math (Do Not Delete) (1)

8/13/2019 Project in Math (Do Not Delete) (1)

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SecondGrading


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Chapter 4:

Polynomial

Functions


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INTRODUCTION:

Linear functions, squarefunctions and cube functions allbelong to the class of functionscalled polynomial functions . Rationalfunctions, on the other hand, areratios of polynomial functions.Evaluating these functions andsolving their equations lead to theFundamental Theorem of Algebra.

The study of graphs ofpolynomial functions is considerableimportance of scientists,astronomers, chemists and physicists

because the properties of thesefunctions affect the behavior andshape of collected data in scientificresearch. For instance, technologyhas its foundations in themathematical concepts ofparabolas and the quadraticformula. They have properties thatare useful in making satellite dishes,car headlights, radio and reflectingtelescopes.

LESSON1: Definition andDegree of PolynomialFunctions

A polynomial function ofdegree n is a function of the formf(x) = a n x n + a n-1 x n-1 + a n-2 x n-2 + +a 0 , where n is a nonnegativeinteger , and a n , a n-1 , a n-2 , , a 0 are real numbers, and a n 0.

The degree of a polynomialfunction is determined by thehighest power of its terms.

1. The polynomial functionf(x) = 5x 3 + 2x 2 + 6 has 3terms.The degrees of the termsare 3, 2, 0 respectively.The highest power of itsterms is 3.Therefore, the degree ofthe polynomial function is3.

2. The polynomial functiong(x) = 4x 2 + 7x 3 + x 4 + 3has 4 terms.The degrees of the terms

are 2, 3, 4 and 0respectively.The highest power of itsterms is 4.Therefore, the degree ofthe polynomial function is4.

LESSON : THE REMAINDER

THEOREM

Remainder Theorem: If apolynomial P(x) is divided by x-c ,where c is a real number, then theremainder is P(c) .

We all know that the onlyoperation in finding the remainderis division . In dividing polynomials,there are 2 sufficient ways ingetting the quotient Q(x). We canget the Q(x) by means of eitherthe two methods: the LongMethod Division or the SyntheticDivision.


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For Long Method Division :

Consider this division:

The quotient here is x 2 x 1, and the remainder is 2. This

result may also be expressed as x 2 x 1 + . This means that thereis a difference of 2 between thedividend x 3 3x 2 + x + 4, and theproduct of the quotient x 2 x 1and the divisor x 2.

Division Algorithm for Polynomials:For each polynomial P(x) of

positive degree n and any realnumber c, there exits a uniquepolynomial Q(x) and a realnumber R such that

P(x) = (x- c) Q(x) + R Where Q(x) is of degree n 1, andR is the remainder.Apply the Remainder Theorem:

x3 3x2 + x + 4 x 2to find the remainder.Solution:

P(x) = x3 3x2 + x + 4Then, x 2 = 0; x = 2. Evaluate.

P(x) = 2 3 3(2 2) + 2 + 4P(x) = 8 12 + 2 + 4P(x) = 2 -- remainder

For convenience, a shortcutmethod for finding the remaindercan be used. It is the syntheticdivision . Using the same problem:

x3 3x 2 + x + 4 x 2

First: Equate the divisor into 0 thentranspose to get the value of x.

x 2 = 0 ; x = 2

Second: Bring down all thenumerical coefficient of the

dividend and place the value of xat the left side.

1 -3 1 4

Third: Bring down the firstnumerical coefficient (which is 1)then multiply it into the divisor.

1 -3 1 4 ____2______

1

Fourth: If they are in the samesign, add the numbers in therespective column. If they haveunlike sign, subtract properly. Thencontinue until you get the lastvalue.

1 -3 1 4 ____2_-2_-2

1 -1 -1 2 - remainder


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Lesson3: finding valuesof polynomial functions

Synthetic division, hand-in-

hand with the RemainderTheorem can be used as aconvenient way to find values ofpolynomial functions.

The Remainder Theoremstates that when the polynomialP(x) is divided by x c, theremainder is P(c).

For example, if thepolynomial P(x) = 2x 3 8x 2 + 19x 12 is divided by x 3, theremainder is P(3).

Nevertheless, finding thevalues of polynomial functions isthe same as what we havelearned in the past lesson. It hasthe same procedure andinstructions.

In summary, the remainder Robtained in synthetic division off(x) by x c, provides theseinformation:

1. The remainder R gives thevalue of f at x = c, that isR = f(c).

2. If R = 0, then x c is a factorof f(x)

3. If R = 0, then (c, 0) is an xintercept of the graph of f.

Lesson4: the factortheorem

FACTOR THEOREM: Let P(x) be a

polynomial. If P(c) = 0, where c isa real number, then x c is afactor of P(x). Conversely, if x c isa factor of P(x), then P(c) = 0.

A. Show that x + 1 is a factor of2x3 + 5x 2 3.

Solution:Let P(x) = 2x 3 + 5x 2 3

P(-1) = 2(-1) 3 + 5(-1) 2 3= -2 + 5 3 = 0

Therefore, x + 1 is a factor of2x3 + 5x 2 3.

If P(c) is not equal to 0, thenthat is not a factor of P(x).

LESSON5: FINDING ZEROES OF

A POLYNOMIAL FUNCTION

With the aid of the theorems just studied the Remainder andthe Factor theorems, factoringtechniques learned and theprocess of synthetic division,finding zeros of a polynomialfunction will be easier.

The zero of a polynomialfunction f(x) is the value of thevariable, which makes thepolynomial function equal to zero.


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A. Solve P(x) = x 3 + 8x 2 + 19x +12, given that one zero is -1.

Solution:By the factor theorem, x+1 is a

factor of x 3 + 8x 2 + 19x + 12. So usethe synthetic division so that youcan have the other zeros. Sincethe highest degree of thepolynomial is 3, we can have 3zeros in P(x).

1 8 19 12-1 -7 -12

1 7 12 0Since you have now the

Q(x) x 2 + 7x + 12, you can simplyuse the four methods in findingthe roots of a quadratic equation.By the Q(x), we can use factoring.

The factors are (x+3)(x+4)respectively. Equate it to zero

then the zeros are -1, -3 and -4 bythe P(x).

Lesson6: rationalzero theorem

Rational Zero Theorem: If f(x) =a n xn + a n-1 xn-1 + + a 1 x + a 0 wherea 0, a 1,, a n are integers (a n 0

and a 0 0) and (in lowest terms)is a rational zero of f(x), then p isan integer factor of the constantterm a 0 and q is an integer factorof the leading coefficient a n .

Illustrative example:Let P(x) = x3 + 6x2 + 10x + 3.

Find the rational zeros of P(x). Ifpossible, find the other zeros.

Solution:By the Rational Zero Theorem, if

is a zero of P(x) then p must be afactor of 3 and q must be a factorof 1.

The possibilities for p are: 1, 3The possibilities for q are: 1.Dividing each p by q, the

resulting possibilities for are:

{ } or simply: 1, 3.Observe that these are the

same as the possibilities for p.Notice that if the coefficient ofthe leading term is 1 (like that inthe example), only the factors ofthe least coefficient is taken aspossibilities of rational zeros.

Notice that all coefficientsare positive. When any positivenumber is substituted by P(x), apositive value is obtained andnever a zero. Therefore, nopositive number can be a rationalzero of P(x). The only possiblezeros are -1 and -3 but P(x) has adegree of 3 and presumably,must have three zeros.

To find other zeros, either

use substitution or syntheticdivision, usually the moreconvenient one.


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Lesson7: graphs of polynomialfunctions

Step1: Find the x- intercepts orzeros of the function.

Recall that you find your x-intercept or zero by setting yourfunction equal to 0, f ( x) = 0,completely factoring the

polynomial and setting each factorequal to 0.

Keep in mind that whenis a factor of your polynomial and

a) if k is even, the graphtouches the x-axis at r andturns around.

b) if k is odd, the graphcrosses the x-axis at r .

Step 2: Find the y -intercept of the

function.

Recall that you can find your y-intercept by letting x = 0 and findyour functional value at x = 0, f(0).

Step 3: Determine if there is anysymmetry.

y -axis symmetry: Recall that your function issymmetric about the y-axis if it isan even function. In other words, if(- x) = f ( x), then your function is

symmetric about the y-axis.

Origin symmetry: Recall that your function issymmetric about the origin if it isan odd function. In other words, if

(- x) = - f ( x), then your function issymmetric about the origin.

Step 4: Find the number ofmaximum turning points.

As discussed above, if f is a polynomial function of degree n,then there is at most n - 1 turning

points on the graph of f .

Step 5: Find extra points, ifneeded.

Sometimes you may need to find points that are in between the onesyou found in steps 2 and 3 to help

you be more accurate on yourgraph.

Step 6: Draw the graph.

Plot the points found in steps 2, 3,and 6 and use the informationgathered in steps 1, 2, 4, and 5 todraw your graph.

Source:http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm


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Chapter summary:

1. A polynomial function of degree n is a function of the form f(x) =a n x n + a n-1 x n-1 + a n-2 x n-2 + + a 0 , where n is a nonnegative integer ,

and a n , a n-1 , a n-2 , , a 0 are real numbers, and a n 0. 2. The degree of a polynomial function is determined by the highestpower of its terms.

3. Division Algorithm for Polynomials: For each polynomial P(x) ofpositive degree n and any real number c, there exists a uniquepolynomial Q(x) and a real number R such thatP(x) = (x- c) Q(x) + R where Q(x) is of degree n -1 and R is theremainder.

4. Remainder Theorem: If a polynomial P(x) is divided by x-c, where c isa real number, then the remainder is P(c).

5. The remainder R obtained in synthetic division of f(x) by x-c, providesthese information:

The remainder R gives the value of f at x = c, that is, R=f(c) If R = 0, then x c is a factor of f(x). If R = 0, then (c, 0) is an x intercept of the graph of f. 6. The zero of a polynomial function is the value of the variable x,

which makes the polynomial function equal to zero. The zero of apolynomial function is also called the root of the correspondingpolynomial equation.

7. Factor Theorem: Let P(x) be a polynomial, if P(c) = 0, where c is areal number, then x c is a factor of P(x). Conversely, if x c is afactor of P(x), then P(c) = 0.

8. The Fundamental Theorem of Algebra states that: Every rationalpolynomial function f(x) = 0 of degree n has exactly n zeros. Also,given any polynomial, if some zeros are given, the remaining zeroscan be found.

Proven by KarlFriedrich Gauss, this fundamental theorem shows that a polynomialfunction of degree n has n complex roots. But, it does not provide a

method or formula for finding a root of any given polynomialequation unlike that for the quadratic equation. For cubic andfourth-degree equations, there are similar but more complicatedformulas. But in 1824, the Norwegian mathematician Niels HenrikAbel (1802 1829 ) showed that no such general formula existsNumerical methods like those described in this chapter are often


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used in finding approximate values for the roots of a polynomialequation.

9. The zeros of any polynomial function correspond to the point ofintersection of the graph of the polynomial function and the x axis.10. Rational Zero Theorem: If a polynomial with integral coefficient hasthe zero , where p and q are relatively prime integers, then p must be

a factor of the constant term of the polynomial, and q, a factor of thecoefficient of the term of highest degree.11. If f is a polynomial function, and r is a real number for which f(r) = 0,

then r is an x intercept of f.12. If (x-r) m is a factor if a polynomial function f, and (x-r)m+1 is not a

factor of f, then r is zero of multiplicity m of f.13. If r is a zero of odd multiplicity, the sign of polynomial function f(x)

changes from one side of r to the other side, and the graph of f(x) crossesthe x axis at r.

If r is a zero of even multiplicity, the sign of polynomial function f(x)remains the same from one side of r to the other side, and the graph off(x) is tangent to the x axis at r.

14. A polynomial function of degree n has n 1 turning points on itsgraph.


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Chapter 5:

Exponentialand

LogarithmicFunctions


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Introduction:

Both exponential andlogarithmic functions are non-

algebraic functions. These arecalled transcendental functionsbecause they cannot beexpressed by a finite number ofalgebraic operations. Anexponential function has aconstant which is raised to avariable power, like f(x) = 2 x, thebasic shape of which whendrawn, indicates a rapidlyincreasing function.Mathematically, the tool thatallows us to operate with suchpower is logarithm. Its laws andproperties can be used to simplifydifficult computations. Althoughmost of its computations havebeen taken over by thecalculator, logarithms continue to

be of basic mathematicalsignificance.The study of these functions

exponential and logarithmic isimportant because of their manyapplications in our contemporarylife. Population growth, growth ofbacterial colonies, decay ofradioactive elements, andcompounded interest offered by

banks are among them. Suchvariations can be represented byexponential equations whichdefine exponential functions.

LESSON1: THE FORM OFEXPONENTIAL FUNCTIONS

Exponential functions are

quite different from otheralgebraic functions, like the linearand the quadratic functions. Theexponential function of the formf(x) = 2 x should not be confusedwith the linear function f(x) = 2x,nor with the polynomial functionf(x) = x 2. As you will observe, thefunction f(x) = 2 x increases rapidly.This is the main feature of manyexponential functions.

LESSON1.1 RELATIONSHIPS WHICH ARE EXPONENTIAL

Remember:

A = P(1 + r) t Where P is the principal, r is the

interest rate compoundedannually, t is the number of years,and A is the amount after t years.

Growth Law: N = N 0e kt Where N 0 is the original populationpresent at time t = 0, N is thepopulation present at time t, k is aconstant (rate of growth) of thepopulation, and e is an irrationalnumber used as base ofexponential function which isapproximately equivalent to2.71828.


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LESSON2: DEFINITION OFEXPONENTIAL FUNCTION

Recall that the table ofvalues consists of ordered pairs ofnumbers (x, f(x)) representing afunction or describing points in acoordinate plane.

Consider the rule f(x) = 2 x forthe geometric progression 2, 4, 8,16, . A table of val ues can begenerated like the one below:

x 0 1 2 3 4 5 6f(x) 1 2 4 8 16 32 64

As you may see that thesequence of numbers in f(x) growsrapidly unlike the growth of alinear and quadratic function.They have the common ratio inf(x) that generates the sequenceof numbers which increases veryrapidly.

Remember: If a > 0 and a 1,then the exponential functionwith base a is the function fdefined by f(x) = a x, where x is anyreal number.

LESSON3: THE GRAPH OF AN

EXPONENTIAL FUNCTION

By drawing the graphs off(x) = a x for different values of a,the characteristics of theexponential function can bedetermined.

LESSON3.1: EXPONENTIAL FUNCTIONOF THE FORM f(x) = AX, where a > 1

From the basic curve, thekey features of an exponentialfunction can be produced:

1. The domain the function f(x)= 2 x is the set of all realnumbers, while the range isthe set of all positive realnumbers.

2. For x > 0, the graph growsvery rapidly.

3. For x < 0, the x axis serves asthe asymptote for thegraph.

An asymptote is a line that acertain part of a graphapproaches more and moreclosely but never reaches.

LESSON3.2: EXPONENTIALFUNCTION OF THE FORM f(x) = ax ,where 0 < a < 1

Consider the graph of f(x) = 2 -x.It can be simplified as f(x) = ( ) x asdefined by the negativeexponents. Notice that they havethe same characteristics as thegraph f(x) = a x shown. The only

difference is the direction. If theexponent is positive, it is anincreasing function. Meanwhile, ifit is negative, it is a decreasingfunction. Meaning to say thegraph f(x) = a x to f(x) = 2 -x is just areflection to each other.

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