[email protected] MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 1 Bruce Mayer, PE Chabot...

[email protected] • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§5.1 Intro to§5.1 Intro toPolyNomialsPolyNomials



Review §Review §

Any QUESTIONS About• §4.3 → Absolute Value: Equations &

InEqualities

Any QUESTIONS About HomeWork• §4.3 → HW-14

4.3 MTH 55



Mathematical “TERMS”Mathematical “TERMS”

A TERM can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables.

A term that is a product of constants and/or variables is called a monomial. • Examples of monomials: 8, w, 24x3y

A polynomial is a monomial or a sum of monomials. Examples of polynomials: • 5w + 8, −3x2 + x + 4, x, 0, 75y6



Example Example Terms Terms

Identify the terms of the polynomial 7p5 − 3p3 + 3

SOLUTION The terms are 7p5, −3p3, and 3.

• We can see this by rewriting all subtractions as additions of opposites:

7p5 − 3p3 + 3 = 7p5 + (−3p3) + 3

These are the terms of the polynomial.



[Bi, Tri, Poly]-nomials[Bi, Tri, Poly]-nomials

A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Polynomials with four or more terms have no special name

Monomials Binomials Trinomials Polynomials

5x2 3x + 4 3x2 + 5x + 9 5x3 6x2 + 2xy 9

8 4a5 + 7bc 7x7 9z3 + 5 a4 + 2a3 a2 + 7a 2

8a23b3 10x3 7 6x2 4x ½ 6x6 4x5 + 2x4 x3 + 3x 2



Polynomial DEGREEPolynomial DEGREE

The degree of a term of a polynomial is the no. of variable factors in that term

EXAMPLE: Determine the degree of each term: a) 9x5 b) 6y c) 9

SOLUTION a) The degree of 9x5 is 5 b) The degree of 6y (6y1) is 1 c) The degree of 9 (9z0) is 0



Mathematical COEFFICIENTMathematical COEFFICIENT The part of a term that is a constant

factor is the coefficient of that term. The coefficient of 4y is 4.

EXAMPLE: Identify the coefficient of each term in polynomial: 5x4 − 8x2y + y − 9

SOLUTION The coefficient of 5x4 is 5. The coefficient of −8x2y is −8. The coefficient of y is 1, since y = 1y. The coefficient of −9 is simply −9



DEGREE of POLYNOMIALDEGREE of POLYNOMIAL

The leading term of a polynomial is the term of highest degree. Its coefficient is called the leading coefficient and its degree is referred to as the degree of the polynomial.

Consider this polynomial4x2 – 9x3 + 6x4 + 8x – 7.

• Find the TERMS, COEFFICIENTS, and DEGREE



DEGREE of POLYNOMIALDEGREE of POLYNOMIAL

For polynomial: 4x2 − 9x3 + 6x4 + 8x − 7• List Terms, CoEfficients, Term-Degree

Terms → 4x2, −9x3, 6x4, 8x, and −7 coefficients → 4, −9, 6, 8 and −7 degree of each term → 2, 3, 4, 1, and 0 The leading term is 6x4 and the

leading coefficient is 6. The degree of the polynomial is 4.



Example Example −−33xx44 + 6 + 6xx33 −− 2 2xx22 + 8 + 8xx + 7 + 7

Term CoefficientTerm

DegreePolyNomial

Degree

–3

6x3

2

1

7

Complete Table for PolyNomial–3x4 + 6x3 – 2x2 + 8x + 7



Example Example ––33xx44 + 6 + 6xx33 – 2 – 2xx22 + 8 + 8xx + 7 + 7


DegreePolyNomial

Degree

–3x4 –3

6x3

–2x2 2

8x 1

7 7

Put Terms in Descending Exponent Order





DegreePolyNomial

Degree

–3x4 −3

6x3 6

–2x2 –2 2

8x 8 1

7 7

Coefficients are the CONSTANTS before the Variables





DegreePolyNomial

Degree

–3x4 –3 4

6x3 6 3

–2x2 –2 2

8x 8 1

7 7 0

Term DEGREE is the Value of the EXPONENT





DegreePolyNomial

Degree

–3x4 –3 4

46x3 6 3

–2x2 –2 2

8x 8 1

7 7 0

Polymomial Degree is the SAME as the highest Term Degree



MultiVariable PolyNomialsMultiVariable PolyNomials

Evaluate the 2-Var polynomial 5 + 4x + xy2 + 9x3y2 for x = −3 & y = 4

Solution: Substitute −3 for x and 4 for y:

5 + 4x + xy2 + 9x3y2

= 5 + 4(−3) + (−3)(4)2 + 9(−3)3(4)2

= 5 − 12 − 48 − 3888

= −3943



Degree of MultiVar PolynomialDegree of MultiVar Polynomial Recall that the degree of a polynomial is the

number of variable factors in the term. Example: ID the coefficient and the degree of each

term and the degree of the polynomial 10x3y2 – 15xy3z4 + yz + 5y + 3x2 + 9

Term Coefficient DegreeDegree of the Polynomial

10x3y2 10 5

8–15xy3z4 –15 8

yz 1 2

5y 5 1

3x2 3 2

9 9 0



Like TermsLike Terms Like, or similar terms either have

exactly the same variables with exactly the same exponents or are constants.

For example,9w5y4 and 15w5y4 are like terms

and −12 and 14 are like terms,

but −6x2y and 9xy3 are not like terms.



Example Example Combine Like Terms Combine Like Terms

a) 10x2y + 4xy3 − 6x2y − 2xy3

b) 8st − 6st2 + 4st2 + 7s3 + 10st − 12s3 + t − 2 SOLUTION

a) 10x2y + 4xy3 − 6x2y − 2xy3

= (10 − 6)x2y + (4 − 2)xy3

= 4x2y + 2xy3

a) 8st − 6st2 + 4st2 + 7s3 + 10st − 12s3+ t − 2

= −5s3 − 2st2 + 18st + t − 2



Common Properties: PolyNom FcnsCommon Properties: PolyNom Fcns

1. The domain of a polynomial function is the set of all real numbers.

2. The graph of a polynomial function is a continuous curve.

• This means that the graph has no holes or gaps and can be drawn on a piece of paper without lifting the pencil.



Continuous vs. DisContinuousContinuous vs. DisContinuous

Could be a PolyNomial Can NOT be

a PolyNomial



Common Properties: PolyNom FcnsCommon Properties: PolyNom Fcns

3. The graph of a polynomial function is a smooth curve.

• This means that the graph of a polynomial function does NOT contain any SHARP corners.



Smooth vs. Kinked/CorneredSmooth vs. Kinked/Cornered

Could be a PolyNomial

Can NOT be a PolyNomial



Leading Coefficient TestLeading Coefficient Test

Given a PolyNomial Function of the form

f x anxn an 1xn 1 ... a1x a0 a 0

The leading term is anxn. The behavior of the graph of f(x) as x → or as x → − is dominated by this term, and is similar to one of the following 4 graphs• Note that The middle portion of each graph,

indicated by the dashed lines, is NOT determined by this test.



Lead Coeff Test: Odd & PositiveLead Coeff Test: Odd & Positive

1. Leading Term• ODD Exponent

• POSITIVE Coeff

Graph• FALLS to LEFT

• RISES to RIGHT

96137 :e.g. 49 xxxf



Lead Coeff Test: Odd & NegativeLead Coeff Test: Odd & Negative

2. Leading Term• ODD Exponent

• NEGATIVE Coeff

Graph• RISES to LEFT

• FALLS to RIGHT

77128 :e.g. 49 xxxf



Lead Coeff Test: Even & PositiveLead Coeff Test: Even & Positive

3. Leading Term• EVEN Exponent

• POSITIVE Coeff

Graph• RISES to LEFT

• RISES to RIGHT

88523 :e.g. 58 xxxf



Lead Coeff Test: Even & NegativeLead Coeff Test: Even & Negative

4. Leading Term• EVEN Exponent

• NEGATIVE Coeff

Graph• FALLS to LEFT

• FALLS to RIGHT 9732 :e.g. 58 xxxf



Example Example Lead CoEff Test Lead CoEff Test

Use the leading-CoEfficient test to determine the end behavior of the graph of

y f x 2x3 3x2 4.

SOLUTION• Here n = 3 (odd) and an = −2 < 0. Thus,

Case-2 (Odd & Neg) applies. The graph of f(x) rises to the left and falls to the right. This behavior is described by: y → as x → −; and y → − as x →



Adding PolynomialsAdding Polynomials

EXAMPLE Add (−6x3 + 7x − 2) + (5x3 + 4x2 + 3)

Solution → Combine Like terms (−6x3 + 7x − 2) + (5x3 + 4x2 + 3)

= (−6 + 5)x3 + 4x2 + 7x + (−2 + 3)

= −x3 + 4x2 + 7x + 1



Example Example Add Polynomials Add Polynomials

Add: (3 – 4x + 2x2) + (–6 + 8x – 4x2 + 2x3)

Solution(3 – 4x + 2x2) + (–6 + 8x – 4x2 + 2x3)

= (3 – 6) + (–4 + 8)x + (2 – 4)x2 + 2x3

= –3 + 4x – 2x2 + 2x3



Example Example Add Polynomials Add Polynomials

Add: 10x5 – 3x3 + 7x2 + 4 and 6x4 – 8x2 + 7 and 4x6 – 6x5 + 2x2 + 6

Solution 10x5 - 3x3 + 7x2 + 4

6x4 - 8x2 + 7

4x6 - 6x5 + 2x2 + 6

4x6 + 4x5 + 6x4 - 3x3 + x2 + 17 Answer: 4x6 + 4x5 + 6x4 − 3x3 + x2 + 17



Opposite of a PolyNomialOpposite of a PolyNomial

To find an equivalent polynomial for the opposite, or additive inverse, of a polynomial, change the sign of every term. • This is the same as multiplying the

original polynomial by −1.



Example Example Opposite of PolyNom Opposite of PolyNom Simplify:

–(–8x4 – x3 + 9x2 – 2x + 72)

Solution–(–8x4 – x3 + 9x2 – 2x + 72)

= (–1)(–8x4 – x3 + 9x2 – 2x + 72)

= 8x4 + x3 – 9x2 + 2x – 72



PolyNomial SubtractionPolyNomial Subtraction

We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted.

PolyNomial Subtractor



Example Example Subtract PolyNom Subtract PolyNom (10x5 + 2x3 – 3x2 + 5) – (–3x5 + 2x4 – 5x3 – 4x2)

Solution (10x5 + 2x3 – 3x2 + 5) – (–3x5 + 2x4 – 5x3 – 4x2)

= 10x5 + 2x3 – 3x2 + 5 + 3x5 – 2x4 + 5x3 + 4x2

= 13x5 – 2x4 + 7x3 + x2 + 5



Example Example Subtract Subtract

(8x5 + 2x3 – 10x) – (4x5 – 5x3 + 6)

Solution (8x5 + 2x3 – 10x) – (4x5 – 5x3 + 6)

= 8x5 + 2x3 – 10x + (–4x5) + 5x3 – 6

= 4x5 + 7x3 – 10x – 6



Example Example Column Form Column Form

Write in columns and subtract:

(6x2 – 4x + 7) – (10x2 – 6x – 4)

Solution

6x2 – 4x + 7

–(10x2 – 6x – 4)

–4x2 + 2x + 11



WhiteBoard WorkWhiteBoard Work

Problems From §5.1 Exercise Set• By ppt → 22, 24, 26, 28, 70

• 10Adding and Subtracting Functions

If f(x) and g(x) define functions, then

(f + g) (x) = f (x) + g(x) Sum function

and (f – g) (x) = f (x) – g(x). Difference function

In each case, the domain of the new function is theintersection of the domains of f(x) and g(x).



P5.1-[22, 24]P5.1-[22, 24]

PolyNomial orNOT PolyNomial

KINKED → NOT a Polynomial

SMOOTH & CONTINUOUS → IS a Polynomial



P5.1-[26, 28]P5.1-[26, 28]

Use Lead CoEfficient Test of End Behavior to Match Fcn to Graph

Odd & Pos → Falls-Lt & Rises-Rt

Odd & Negs → Rise-Lt & Falls-Rt



P5.1-70 P5.1-70 AIDS Mortality Models AIDS Mortality Models Given PolyNomial Models for USA

AIDS mortality over the years 1990-2002 where x ≡ yrs since 1990

111568549251844 2 xxxf

11059056036206611 23 xxxxg Bar Chart shows ACTUAL 2002

Mortality of 501 669 Find Error Associated with Each Model



P5.1-70 P5.1-70 AIDS Mortality Models AIDS Mortality Models Evaluate Model

using MATLAB Math-Processing Software • See MTH25 for

Info on MATLAB

>> x =2002-1990x = 12

>> fx = -1844*x^2 + 54923*x + 111568fx = 505108

>> gx = -11*x^3 - 2066*x^2 + 56036*x + 110590

gx = 466510

>> Yactual = 501669

>> fx_error = (fx-Yactual)/Yactualfx_error = 0.0069 = 0.69%

>> gx_error = (gx-Yactual)/Yactualgx_error = -0.0701 = -7.01%

By MATLAB the Model Errors• f(x) → 0.69% Low

• g(x) → 7.0% Low



All Done for TodayAll Done for Today

Lead CoeffTest

Summarized

n is Even

an > 0

n is Even

an < 0

n is Odd

an > 0

n is Odd

an < 0



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



Graph Graph yy = | = |xx||

Make T-tablex y = |x |

-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

x

y

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



x

y

-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls

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