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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
[email protected]
Chabot Mathematics
§5.1 Intro to§5.1 Intro toPolyNomialsPolyNomials
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §4.3 → Absolute Value: Equations &
InEqualities
Any QUESTIONS About HomeWork• §4.3 → HW-14
4.3 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
[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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example ––33xx44 + 6 + 6xx33 – 2 – 2xx22 + 8 + 8xx + 7 + 7
Term CoefficientTerm
DegreePolyNomial
Degree
–3x4 –3
6x3
–2x2 2
8x 1
7 7
Put Terms in Descending Exponent Order
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Bruce Mayer, PE Chabot College Mathematics
Example Example ––33xx44 + 6 + 6xx33 – 2 – 2xx22 + 8 + 8xx + 7 + 7
Term CoefficientTerm
DegreePolyNomial
Degree
–3x4 −3
6x3 6
–2x2 –2 2
8x 8 1
7 7
Coefficients are the CONSTANTS before the Variables
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Bruce Mayer, PE Chabot College Mathematics
Example Example ––33xx44 + 6 + 6xx33 – 2 – 2xx22 + 8 + 8xx + 7 + 7
Term CoefficientTerm
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example ––33xx44 + 6 + 6xx33 – 2 – 2xx22 + 8 + 8xx + 7 + 7
Term CoefficientTerm
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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.
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Continuous vs. DisContinuousContinuous vs. DisContinuous
Could be a PolyNomial Can NOT be
a PolyNomial
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
Smooth vs. Kinked/CorneredSmooth vs. Kinked/Cornered
Could be a PolyNomial
Can NOT be a PolyNomial
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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 →
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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.
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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
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Bruce Mayer, PE Chabot College Mathematics
PolyNomial SubtractionPolyNomial Subtraction
We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted.
PolyNomial Subtractor
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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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).
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P5.1-[22, 24]P5.1-[22, 24]
PolyNomial orNOT PolyNomial
KINKED → NOT a Polynomial
SMOOTH & CONTINUOUS → IS a Polynomial
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
[email protected]
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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