Section 5.1 Polynomial Functions and Models Term: A term is an expression that involves only multiplication and/or division with constants and/or variables. A term is separated by ‘+’ or ‘–’ Polynomial: A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents A polynomial is a monomial or the indicated sum or difference of monomials A Monomial is a polynomial that has exactly one term A Binomial is a polynomial with two terms A Trinomial is a polynomial with three terms Coefficient: A number written next to a variable indicates multiplication. Leading Coefficient is the coefficient of the term of the largest degree Degree of a Polynomial is the greatest degree of all the terms of the polynomial Degree of a nonzero constant term is 0 The constant ‘0’ has no defined degree. Examples Number Types Degree Leading Expression Of Of Of Coefficient Terms Polynomial Polynomial 2 1 Monomial 0 –2 1 Monomial 0 2x 1 Monomial 1 2 –2x 1 Monomial 1 –2 –53x3 1 Monomial 3 –53 4x + x3 2 Binomial 3 1 4 2 Binomial 4 –1 5 7 3 Trinomial 3 5 1 2 Not Polynomial since the exponent is not whole # 3 4 2 Not Polynomial since the exponent 2 is not whole # 2 3 Not Polynomial since the exponent is not whole # 3 2 1 3 0 1 Monomial Undefined Definition of a Polynomial function: A polynomial function is a function that can be written in the form of ⋯ 0 , where , , , are real numbers and is nonnegative integer. The number is the leading coefficient, the coefficient of the variable to the highest power. The degree of the polynomial function is . Cheon-Sig Lee Page 1 Section 5.1 Polynomial Functions and Models End Behavior of Polynomial Functions ⋯ 0 If the Degree of a polynomial function is Odd, , leading coefficient, is POSITIVE , leading coefficient, is NEGATIVE The graph falls left and rises right The graph rises left and falls right If the Degree of a polynomial function is Even, , leading coefficient, is POSITIVE , leading coefficient, is NEGATIVE The graph rises left and right The graph falls left and right Power Functions A power function of degree is a monomial function of the form , where is a none-zero real number and is a positive integer. Turning Points If is a polynomial function of degree , then has at most 1 turning points If the graph of has 1 turning points, the degree of is as least . Zeros of Polynomial Functions The zero of polynomial function is the values of x for which is equal to 0. If is a function of a real number for which , then is called a real zero of . The following statements are equivalent is a real zero of a polynomial function . is an x-intercept of the graph of . is a solution to the equation 0. is a factor of . Multiplicity Multiplicities of Zeros Zero If is even, then the graph touches the x-axis and turns around at . If is odd, then the graph crosses the x-axis at . Cheon-Sig Lee Page 2 Section 5.1 Polynomial Functions and Models Real Zeros and Factors of a function For a real number leading coefficient and real zeros ,,⋯, of a function If the multiplicities of real zeros are , ,⋯, respectively, then the factor form of the function is ⋯ Finding Zeros of Polynomial Functions Method 1: Factoring by hand to find zeros Method 2: Using a graphing calculator to find zeros of polynomial functions. TI-83: Y= > ⋯ > 2ndMODE > 2ndTRACE > 2: ZERO TI-83: Y= > ⋯ > 2nd GRAPH TI-89: F2 > 2: Solve ⋯ 0, Intermediate Value Theorem for Polynomial Functions Let be a polynomial function with real number coefficients. If and have opposite signs, then there is at least one value of between and for which 0. Exercises 1. (Solution 1) The factor form of a polynomial is ⋯ The leading coefficient 1. Real zeros 1, 0, 2. 1 because the number of zeros is equal to the degree of the polynomial. 11 02 12 2. (Solution 2) The factor form of a polynomial is ⋯ 1, 2, 1, 3, 4 1 because the number of zeros is equal to the degree of the polynomial. 12 1 34 2134 3. (Solution 3) The factor form of a polynomial is ⋯ 1, 9, 1, 3, 2 193 93 Cheon-Sig Lee Page 3 Section 5.1 Polynomial Functions and Models 4. (Solution 4.a) If is a factor of a polynomial function , is a zero of multiplicity of . 8 0→0⇒ the zero is 8 and the multiplicity of 8 is 1. 4 0→4⇒ the zero is 4 and the multiplicity of 4 is 2. (Solution 4.b) Because the larger zero is 8 and its multiplicity is odd, the graph crosses the x-axis at the larger x-intercept. Because the smaller zero is 4 and its multiplicity is even, the graph touches the x-axis at the smaller x-intercept. (Solution 4.c) The maximum number of turning points is one less than the degree of the polynomial. Because the degree of the function is 3, the maximum number of turning points is 312 (Solution 4.d) The graph of the polynomial ⋯ resembles the graph of . The degree of 784 is 3. Thus, its graph resembles the graph of 7. 5. (Solution 5.a) 49 0→ the solution to 490 is not real number 2 0→2⇒ the zero is 2 and its multiplicity is 3 Cheon-Sig Lee Page 4 Section 5.1 Polynomial Functions and Models Question 5 continued… (Solution 5.b) Because the multiplicity of 2 is odd, the graph crosses the x-axis at the larger x-intercept. (Solution 5.c) 49 → Degree is 2; 2 → Degree is 3. Therefore, 8 492 → Degree is 5 Because the degree of the function is 5, the maximum number of turning points is 514. (Solution 5.c) The degree of 8 492 is 5. Thus, its graph resembles the graph of 8. 6. (Solution 6.a) 0→0⇒ the zero is 0 and its multiplicity is 2. 7 0→√7 √70⇒√7, √7 ⇒ zeros are √7 and √7 . The multiplicity of √7 is 1. The multiplicity of √7 is 1. (Solution 6.b) Because the largest x-intercept is √7 whose multiplicity is odd, the graph crosses the x-axis at the point. Because the middle x-intercept is 0 whose multiplicity is even, the graph touches the x-axis at the point. Because the smallest x-intercept is √7 whose multiplicity is odd, the graph crosses the x-axis at the point. (Solution 6.c) → Degree is 2; 7 → Degree is 2. Therefore, 7 7 → Degree is 4 Because the degree of the function is 5, the maximum number of turning points is 413. (Solution 6.d) The degree of 7 7 is 4. Thus, its graph resembles the graph of 7 Cheon-Sig Lee Page 5 .