MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §5.1 Intro to PolyNomials

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.3 → Absolute Value: Equations & InEqualities  Any QUESTIONS About HomeWork §4.3 → HW MTH 55

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 3 Bruce Mayer, PE Chabot College Mathematics 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, 24x 3 y  A polynomial is a monomial or a sum of monomials. Examples of polynomials: 5w + 8, −3x 2 + x + 4, x, 0, 75y 6

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Terms  Identify the terms of the polynomial 7p 5 − 3p  SOLUTION  The terms are 7p 5, −3p 3, and 3. We can see this by rewriting all subtractions as additions of opposites: 7p 5 − 3p = 7p 5 + (−3p 3 ) + 3 These are the terms of the polynomial.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 5 Bruce Mayer, PE Chabot College Mathematics [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 MonomialsBinomialsTrinomialsPolynomials 5x25x2 3x + 43x 2 + 5x + 9 5x 3  6x 2 + 2xy  9 84a 5 + 7bc 7x 7  9z 3 + 5a 4 + 2a 3  a 2 + 7a  2  8a 23 b 3  10x 3  76x 2  4x  ½6x 6  4x 5 + 2x 4  x 3 + 3x  2

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 6 Bruce Mayer, PE Chabot College Mathematics Polynomial 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) 9x 5 b) 6y c) 9  SOLUTION  a) The degree of 9x 5 is 5  b) The degree of 6y (6y 1 ) is 1  c) The degree of 9 (9z 0 ) is 0

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 7 Bruce Mayer, PE Chabot College Mathematics Mathematical 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: 5x 4 − 8x 2 y + y − 9  SOLUTION  The coefficient of 5x 4 is 5.  The coefficient of −8x 2 y is −8.  The coefficient of y is 1, since y = 1y.  The coefficient of −9 is simply −9

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 8 Bruce Mayer, PE Chabot College Mathematics DEGREE 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 polynomial 4x 2 – 9x 3 + 6x 4 + 8x – 7. Find the TERMS, COEFFICIENTS, and DEGREE

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 9 Bruce Mayer, PE Chabot College Mathematics DEGREE of POLYNOMIAL  For polynomial: 4x 2 − 9x 3 + 6x 4 + 8x − 7 List Terms, CoEfficients, Term-Degree  Terms → 4x 2, −9x 3, 6x 4, 8x, and −7  coefficients → 4, −9, 6, 8 and −7  degree of each term → 2, 3, 4, 1, and 0  The leading term is 6x 4 and the leading coefficient is 6.  The degree of the polynomial is 4.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  −3x 4 + 6x 3 − 2x 2 + 8x + 7 TermCoefficient Term Degree PolyNomial Degree –3 6x  Complete Table for PolyNomial –3x 4 + 6x 3 – 2x 2 + 8x + 7

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  –3x 4 + 6x 3 – 2x 2 + 8x + 7 TermCoefficient Term Degree PolyNomial Degree –3x 4 –3 6x 3 –2x 2 2 8x  Put Terms in Descending Exponent Order

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  –3x 4 + 6x 3 – 2x 2 + 8x + 7 TermCoefficient Term Degree PolyNomial Degree –3x 4 −3 6x 3 6 –2x 2 –2 2 8x  Coefficients are the CONSTANTS before the Variables

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  –3x 4 + 6x 3 – 2x 2 + 8x + 7 TermCoefficient Term Degree PolyNomial Degree –3x 4 –34 6x –2x 2 –2 2 8x  Term DEGREE is the Value of the EXPONENT

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  –3x 4 + 6x 3 – 2x 2 + 8x + 7 TermCoefficient Term Degree PolyNomial Degree –3x 4 –34 4 6x –2x 2 –2 2 8x  Polymomial Degree is the SAME as the highest Term Degree

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 15 Bruce Mayer, PE Chabot College Mathematics MultiVariable PolyNomials  Evaluate the 2-Var polynomial 5 + 4x + xy 2 + 9x 3 y 2 for x = −3 & y = 4  Solution: Substitute −3 for x and 4 for y: 5 + 4x + xy 2 + 9x 3 y 2 = 5 + 4(−3) + (−3)(4) 2 + 9(−3) 3 (4) 2 = 5 − 12 − 48 − 3888 = −3943

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 16 Bruce Mayer, PE Chabot College Mathematics Degree 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 10x 3 y 2 – 15xy 3 z 4 + yz + 5y + 3x TermCoefficientDegree Degree of the Polynomial 10x 3 y –15xy 3 z 4 –158 yz12 5y5y51 3x23x

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 17 Bruce Mayer, PE Chabot College Mathematics Like Terms  Like, or similar terms either have exactly the same variables with exactly the same exponents or are constants.  For example, 9w 5 y 4 and 15w 5 y 4 are like terms  and −12 and 14 are like terms,  but −6x 2 y and 9xy 3 are not like terms.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Combine Like Terms a)10x 2 y + 4xy 3 − 6x 2 y − 2xy 3 b)8st − 6st 2 + 4st 2 + 7s st − 12s 3 + t − 2  SOLUTION a)10x 2 y + 4xy 3 − 6x 2 y − 2xy 3 = (10 − 6)x 2 y + (4 − 2)xy 3 = 4x 2 y + 2xy 3 b)8st − 6st 2 + 4st 2 + 7s st − 12s 3 + t − 2 = −5s 3 − 2st st + t − 2

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 19 Bruce Mayer, PE Chabot College Mathematics Common 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.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 20 Bruce Mayer, PE Chabot College Mathematics Continuous vs. DisContinuous Could be a PolyNomial Can NOT be a PolyNomial

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 21 Bruce Mayer, PE Chabot College Mathematics Common 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.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 22 Bruce Mayer, PE Chabot College Mathematics Smooth vs. Kinked/Cornered Could be a PolyNomial Can NOT be a PolyNomial

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 23 Bruce Mayer, PE Chabot College Mathematics Leading Coefficient Test  Given a PolyNomial Function of the form  The leading term is a n x n. 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.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 24 Bruce Mayer, PE Chabot College Mathematics Lead Coeff Test: Odd & Positive 1.Leading Term ODD Exponent POSITIVE Coeff  Graph FALLS to LEFT RISES to RIGHT

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 25 Bruce Mayer, PE Chabot College Mathematics Lead Coeff Test: Odd & Negative 2.Leading Term ODD Exponent NEGATIVE Coeff  Graph RISES to LEFT FALLS to RIGHT

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 26 Bruce Mayer, PE Chabot College Mathematics Lead Coeff Test: Even & Positive 3.Leading Term EVEN Exponent POSITIVE Coeff  Graph RISES to LEFT RISES to RIGHT

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 27 Bruce Mayer, PE Chabot College Mathematics Lead Coeff Test: Even & Negative 4.Leading Term EVEN Exponent NEGATIVE Coeff  Graph FALLS to LEFT FALLS to RIGHT

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Lead CoEff Test  Use the leading-CoEfficient test to determine the end behavior of the graph of  SOLUTION Here n = 3 (odd) and a n = −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 → 

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 29 Bruce Mayer, PE Chabot College Mathematics Adding Polynomials  EXAMPLE  Add (−6x 3 + 7x − 2) + (5x 3 + 4x 2 + 3)  Solution → Combine Like terms (−6x 3 + 7x − 2) + (5x 3 + 4x 2 + 3) = (−6 + 5)x 3 + 4x 2 + 7x + (−2 + 3) = −x 3 + 4x 2 + 7x + 1

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Add Polynomials  Add: (3 – 4x + 2x 2 ) + (–6 + 8x – 4x 2 + 2x 3 )  Solution (3 – 4x + 2x 2 ) + (–6 + 8x – 4x 2 + 2x 3 ) = (3 – 6) + (–4 + 8)x + (2 – 4)x 2 + 2x 3 = –3 + 4x – 2x 2 + 2x 3

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Add Polynomials  Add: 10x 5 – 3x 3 + 7x and 6x 4 – 8x and 4x 6 – 6x 5 + 2x  Solution 10x 5 - 3x 3 + 7x x 4 - 8x x 6 - 6x 5 + 2x x 6 + 4x 5 + 6x 4 - 3x 3 + x  Answer: 4x 6 + 4x 5 + 6x 4 − 3x 3 + x

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 32 Bruce Mayer, PE Chabot College Mathematics Opposite 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.

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Opposite of PolyNom  Simplify: –(–8x 4 – x 3 + 9x 2 – 2x + 72)  Solution –(–8x 4 – x 3 + 9x 2 – 2x + 72) = (–1)(–8x 4 – x 3 + 9x 2 – 2x + 72) = 8x 4 + x 3 – 9x 2 + 2x – 72

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 34 Bruce Mayer, PE Chabot College Mathematics PolyNomial Subtraction  We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted. PolyNomial Subtractor

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Subtract PolyNom  (10x 5 + 2x 3 – 3x 2 + 5) – (–3x 5 + 2x 4 – 5x 3 – 4x 2 )  Solution (10x 5 + 2x 3 – 3x 2 + 5) – (–3x 5 + 2x 4 – 5x 3 – 4x 2 ) = 10x 5 + 2x 3 – 3x x 5 – 2x 4 + 5x 3 + 4x 2 = 13x 5 – 2x 4 + 7x 3 + x 2 + 5

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  Subtract  (8x 5 + 2x 3 – 10x) – (4x 5 – 5x 3 + 6)  Solution (8x 5 + 2x 3 – 10x) – (4x 5 – 5x 3 + 6) = 8x 5 + 2x 3 – 10x + (–4x 5 ) + 5x 3 – 6 = 4x 5 + 7x 3 – 10x – 6

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Column Form  Write in columns and subtract: (6x 2 – 4x + 7) – (10x 2 – 6x – 4)  Solution 6x 2 – 4x + 7 –(10x 2 – 6x – 4) –4x 2 + 2x + 11

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 38 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.1 Exercise Set By ppt → 22, 24, 26, 28, Adding and Subtracting Functions If f ( x ) and g ( x ) define functions, then ( f + g ) ( x ) = f + g(x)g(x)Sum function and( f – g ) ( x ) = f – g ( x ).Difference function In each case, the domain of the new function is the intersection of the domains of f ( x ) and g ( x ).

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 39 Bruce Mayer, PE Chabot College Mathematics P5.1-[22, 24]  PolyNomial or NOT PolyNomial KINKED → NOT a Polynomial SMOOTH & CONTINUOUS → IS a Polynomial

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 40 Bruce Mayer, PE Chabot College Mathematics 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

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 41 Bruce Mayer, PE Chabot College Mathematics P  AIDS Mortality Models  Given PolyNomial Models for USA AIDS mortality over the years where x ≡ yrs since 1990  Bar Chart shows ACTUAL 2002 Mortality of  Find Error Associated with Each Model

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 42 Bruce Mayer, PE Chabot College Mathematics P  AIDS Mortality Models  Evaluate Model using MATLAB Math-Processing Software See MTH25 for Info on MATLAB >> x = x = 12 >> fx = -1844*x^ *x fx = >> gx = -11*x^ *x^ *x gx = >> Yactual = >> fx_error = (fx-Yactual)/Yactual fx_error = = 0.69% >> gx_error = (gx-Yactual)/Yactual gx_error = = -7.01%  By MATLAB the Model Errors f(x) → 0.69% Low g(x) → 7.0% Low

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 43 Bruce Mayer, PE Chabot College Mathematics All Done for Today Lead Coeff Test Summarized n is Even a n > 0 n is Even a n < 0 n is Odd a n > 0 n is Odd a n < 0

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 44 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 45 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt 46 Bruce Mayer, PE Chabot College Mathematics