CHAPTER Polnomial Functions Section. Quadratic Functions and Models............. 7 Section. Polnomial Functions of Higher Degree......... 7 Section. Polnomial and Snthetic Division............ Section. Zeros of Polnomial Functions.............. 9 Section. Mathematical Modeling and Variation.......... Review Eercises............................ Problem Solving............................. Practice Test..............................
CHAPTER Polnomial Functions Section. Quadratic Functions and Models You should know the following facts about parabolas. f a b c, a, is a quadratic function, and its graph is a parabola. If a >, the parabola opens upward and the verte is the point with the minimum -value. If a <, the parabola opens downward and the verte is the point with the maimum -value. The verte is ba, f ba. To find the -intercepts (if an), solve a b c. The standard form of the equation of a parabola is f a h k where a. (a) The verte is h, k. (b) The ais is the vertical line h. Vocabular Check. nonnegative integer; real. quadratic; parabola. ais or ais of smmetr. positive; minimum. negative; maimum. f opens upward and has verte,.. f opens upward and has verte,. Matches graph (g). Matches graph (b).. f opens downward 7. f opens downward and has and has verte,. Matches graph (f). verte,. Matches graph (e). 9. (a) (b) Vertical shrink Vertical shrink and reflection in the -ais CONTINUED 7
Chapter Polnomial Functions 9. CONTINUED (d) Vertical stretch Vertical stretch and reflection in the -ais. (a) (b) Horizontal translation one unit to the right Horizontal shrink and a vertical translation one unit upward (d) Horizontal stretch and a vertical translation three units downward Horizontal translation three units to the left. f. f Verte:, Verte:, Ais of smmetr: Find -intercepts: ± -intercepts:,,, or the -ais Ais of smmetr: Find -intercepts: -intercepts: ± ±,,, or the -ais
Section. Quadratic Functions and Models 9 7. f 9. Verte:, Ais of smmetr: Find -intercepts: ± ± -intercepts:,,, h Verte:, Ais of smmetr: -intercept:,. f. f Verte:, Verte:, Ais of smmetr: Find -intercepts: Ais of smmetr: Find -intercepts: ± Not a real number ± -intercepts: ±,,, No -intercepts. h Verte:, Ais of smmetr: Find -intercepts: Not a real number ± No -intercepts
7 Chapter Polnomial Functions 7. f 9. f Verte:, Verte:, Ais of smmetr: -intercepts:,,, 7 Ais of smmetr: Find -intercepts: or -intercepts:,,,. g. f Verte:, Ais of smmetr: -intercepts: ±, Verte:, Ais of smmetr: -intercepts: ±,. g 7., is the verte. Verte:, a a Ais of smmetr: -intercepts: ±, Since the graph passes through the point,, we have: a a 9., is the verte.., is the verte. a Since the graph passes through the point,, we have: a a a a Since the graph passes through the point,, we have: a a., is the verte.., is the verte. f a Since the graph passes through the point, 9, we have: 9 a a a f f a Since the graph passes through the point,, we have: a a a f
7., is the verte. 9. is the verte. f a Since the graph passes through the point 7,, we have: a7 a a f Section. Quadratic Functions and Models 7, f a Since the graph passes through the point we have: a 9 a a 9 f 9,,., is the verte.. f a Since the graph passes through the point we have: a7 a f 7,, -intercepts: ±, ±. 7. -intercepts:,,, or f -intercepts:,, (, ) or The -intercepts and the solutions of f are the same. 9. f 9. f 7 -intercepts:,,, -intercepts:,,, 9 ) or The -intercepts and the solutions of f are the same. 7 ) or The -intercepts and the solutions of f are the same.. f 7. f opens upward -intercepts:,, 7, 7 7 7 or 7 The -intercepts and the solutions of f are the same. g opens downward Note: f a has -intercepts, and, for all real numbers a.
7 Chapter Polnomial Functions 7. f opens upward 9. f opens upward g opens downward Note: f a a has -intercepts, and, for all real numbers a. 7 g 7 7 opens downward Note: f a has -intercepts, and for all real numbers a., 7. Let the first number and the second number. 7. Let the first number and the second number. Then the sum is Then the sum is. The product is P. P The maimum value of the product occurs at the verte of P and is. This happens when. The product is P. P. 7 The maimum value of the product occurs at the verte of P and is 7. This happens when and. Thus, the numbers are and. 7. (a) (b) A This area is maimum when feet and feet. A This area is maimum when feet and feet. CONTINUED
Section. Quadratic Functions and Models 7 7. CONTINUED (d) A (e) The are all identical. feet and feet The maimum area occurs at the verte and is square feet. This happens when feet and feet. The dimensions are feet b feet. 77. 9 9 The verte occurs at b 9. The maimum height is feet. 9 a 9 9 79. C.. The verte occurs at b. The cost is minimum when fitures. a.. P., The verte occurs at b The profit is maimum when, units. a,... Rp p p. C 99.t.t, t (a) R $, thousand R $,7 thousand R $, thousand (b) The revenue is a maimum at the verte b a R, The unit price that will ield a maimum revenue of $, thousand is $. (a) (b) Verte, 99 The verte occurs when 99 which is the maimum average annual consumption. The warnings ma not have had an immediate effect, but over time the and other findings about the health risks and the increased cost of cigarettes have had an effect. C Annuall: Dail: 79 9,,9,,9 cigarettes 79 cigarettes
7 Chapter Polnomial Functions 7. (a) 9. True. The equation has no real solution, so the graph has no -intercepts. (b).s.s.9 a, b, c,9 s ±,9 s s s,9 ±,7 s 7., 9. s s 9, The maimum speed if power is not to eceed horsepower is 9. miles per hour. 9. f a b c a b a c a b a b a a b a b a c a a b ac b a f b a a b a b b a c b b a a c b b b ac a a c So, the verte occurs at b ac b, a a ac b a b a, f b a. 9. Yes. A graph of a quadratic equation whose verte is, has onl one -intercept. 9., and, 97. and m m The slope of the perpendicular line through, is m and the -intercept is b.
Section. Polnomial Functions of Higher Degree 7 For Eercises 99, let f, and g. 99. f g f g. 7 fg 7 f 7 g 7 7 7 9 9. f g fg f 9 Section. Polnomial Functions of Higher Degree You should know the following basic principles about polnomials. f a n n a n n... a a a, a n, is a polnomial function of degree n. If f is of odd degree and (a) a n >, then (b) a n <, then. f as.. f as.. f as.. f as. If f is of even degree and (a) a n >, then (b) a n <, then. f as.. f as.. f as.. f as. The following are equivalent for a polnomial function. (a) a is a zero of a function. (b) a is a solution of the polnomial equation f. a is a factor of the polnomial. (d) a, is an -intercept of the graph of f. A polnomial of degree n has at most n distinct zeros and at most n turning points. A factor a k, k >, ields a repeated zero of a of multiplicit k. (a) If k is odd, the graph crosses the -ais at a. (b) If k is even, the graph just touches the -ais at a. If f is a polnomial function such that a < b and fa fb, then f takes on ever value between fa and fb in the interval a, b. If ou can find a value where a polnomial is positive and another value where it is negative, then there is at least one real zero between the values. Vocabular Check. continuous. Leading Coefficient Test. n; n. solution; a; -intercept. touches; crosses. standard 7. Intermediate Value
7 Chapter Polnomial Functions. f is a line with -intercept,. Matches. f is a parabola with -intercepts graph., and, and opens downward. Matches graph (h).. f has intercepts, and ±,. 7. f has intercepts, and,. Matches graph (a). Matches graph (d). 9. (a) f (b) f Horizontal shift two units to the right Vertical shift two units downward f (d) f Reflection in the -ais and a vertical shrink Horizontal shift two units to the right and a vertical shift two units downward. (a) f (b) f Horizontal shift three units to the left Vertical shift three units downward CONTINUED
Section. Polnomial Functions of Higher Degree 77. CONTINUED f (d) f (e) Reflection in the -ais and then a vertical shift four units upward f (f) Horizontal shift one unit to the right and a vertical shrink each -value is multiplied b f Vertical shift one unit upward and a horizontal shrink each -value is multiplied b Vertical shift two units downward and a horizontal stretch each -value is multiplied b. f. g 7 Degree: Degree: Leading coefficient: Leading coefficient: The degree is odd and the leading coefficient is positive. The graph falls to the left and rises to the right. The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. 7. f. 9. f Degree: Leading coefficient:. Degree: Leading coefficient: The degree is odd and the leading coefficient is negative. The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. The graph rises to the left and falls to the right.. ht t t. f 9 ; g Degree: Leading coefficient: The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. g f
7 Chapter Polnomial Functions. f ; g 7. g f f (a) Zeros: ± (b) Each zero has a multiplicit of. (odd multiplicit) Turning point: (the verte of the parabola) 9. ht t t 9. (a) t t 9 t Zero: t (b) t has a multiplicit of (even multiplicit). Turning point: (the verte of the parabola) f (a) Zeros:, (b) Each zero has a multiplicit of (odd multiplicit). Turning point: (the verte of the parabola). f. (a) Zeros:, ± (b the Quadratic Formula) (b) Each zero has a multiplicit of (odd multiplicit). Turning points: ft t t t (a) (b) t t t tt t tt Zeros: t, t t has a multiplicit of (odd multiplicit) t has a multiplicit of (even multiplicit) Turning points: 7 7. gt t t 9t (a) t t 9t tt t 9 tt tt t Zeros: t, t ± 9 9 (b) t has a multiplicit of (odd multiplicit). t ± each have a multiplicit of (even multiplicit) Turning points:
Section. Polnomial Functions of Higher Degree 79 9. f. (a) No real zeros (b) Turning point: g (a) Zeros: ±, (b) Each zero has a multiplicit of (odd multiplicit). Turning points: 7.. (a) (a) (b) -intercepts: or,,, (d) The solutions are the same as the -coordinates of the -intercepts. (b) -intercepts:,, ±,, ±,, ±, ± (d) The solutions are the same as the -coordinates of the -intercepts. 7. f 9. f Note: f a a has zeros and for all real numbers a. f f f Note: f a has zeros and for all real numbers a.. f. Note: f a has zeros,, for all real numbers a. f 9 9 Note: f a 9 has these zeros for all real numbers a.. f 7. Note: f a has these zeros for all real numbers a. f Note: f a, a, has degree and zero.
Chapter Polnomial Functions 9. f. Note: f a, a, has degree and zeros,,. f Note: f a, a, has degree and zeros,,.. f 7 or f or f 7 Note: An nonzero scalar multiple of these functions would also have degree and zeros,,.. f or f or f or f 9 Note: An nonzero scalar multiple of these functions would also have degree and zeros and. 7. f 9 9 9. f t t t t 7 (a) Falls to the left; rises to the right (a) Rises to the left; rises to the right (b) Zeros:,, (b) No real zero (no -intercepts) t f f t..7..7. (d) (d) The graph is a parabola with verte, 7. (, ) (, ) (, ) t 7. f 7. f (a) Falls to the left; rises to the right (a) Falls to the left; rises to the right (b) Zeros: and (b) Zeros:,,.. f f.7 7.7 (d) (d) (, ) (, ) 7 (, ) (, ) (, )
Section. Polnomial Functions of Higher Degree 7. f 77. f (a) Rises to the left; falls to the right (a) Falls to the left; rises to the right (b) Zeros:, (b) Zeros:, f f 9 (d) (d) (, ) (, ) (, ) (, ) 79. gt t t. f (a) Falls to the left; falls to the right (b) Zeros: and t 9 9 (d) gt (, ) (, ) 9 9 t Zeros:,, all of multiplicit. g 9. f Zeros: of multiplicit, of multiplicit, and of multiplicit 9 The function has three zeros. The are in the intervals,,, and,. The are.79,.7,.. 7 9
Chapter Polnomial Functions 7. g 9 The function has two zeros. The are in the intervals, and,. The are.,.779. 77 9. (a) Thus, V. (b) Domain: (d) Volume l w h height length width The length and width must be positive. < < The maimum point on the graph occurs at. This agrees with the maimum found in part. Bo Bo Bo Height Width Volume, V 7 7 7 7 7 The volume is a maimum of cubic inches when the height is inches and the length and width are each inches. So the dimensions are inches. 9. (a) A w square inches (b) feet 9 inches V l w h 9 cubic inches Since and cannot be negative, we have < < inches for the domain. (d) When, the volume is a V maimum with V in.. The 9 dimensions of the gutter cross-section are inches inches inches. (e) Maimum:, The maimum value is the same. (f) No. The volume is a product of the constant length and the cross-sectional area. The value of would remain the same; onl the value of V would change if the length was changed. 7 7 9
Section. Polnomial Functions of Higher Degree 9..9t.t.t 9 7 The model is a good fit to the actual data. 9. Midwest: South: $9. thousand $9, $.7 thousand $,7 Since the models are both cubic functions with positive leading coefficients, both will increase without bound as t increases, thus should onl be used for short term projections. 97. G.t.7t.t.9, t (a) (b) The tree is growing most rapidl at t..9t.7t. b a.7..9.. Verte.,. (d) The -value of the verte in part is approimatel equal to the value found in part (b). 99. False. A fifth degree polnomial can have at most four turning points.. True. A polnomial of degree 7 with a negative leading coefficient rises to the left and falls to the right.. f ; f is even. (a) (d) g f (f) g f Vertical shift two units upward g f Even (b) g f Horizontal shift two units to the left Neither odd nor even Reflection in the -ais Even g f Vertical shrink Even (h) g f f f f f Even f g (e) g f Reflection in the -ais. The graph looks the same. Even g f Horizontal stretch Even (g) g f, Neither odd nor even. 7 7. 7 7
Chapter Polnomial Functions 9.. 7 7 7.. ± ± ± ± 7. f 9. f Common function: Transformation: Horizontal shift four units to the left 7 7 Common function: Transformation: Horizontal shift one unit to the left and a vertical shift five units downward. f 9 Common function: Transformation: Vertical stretch each -value is multiplied b, then a vertical shift nine units upward Section. Polnomial and Snthetic Division You should know the following basic techniques and principles of polnomial division. The Division Algorithm (Long Division of Polnomials) Snthetic Division f k is equal to the remainder of f divided b k. (The Remainder Theorem) fk if and onl if k is a factor of f.
Section. Polnomial and Snthetic Division Vocabular Check. f is the dividend; d is the divisor; g is the quotient; r is the remainder. improper; proper. snthetic division. factor. remainder. and ) Thus, and.. and (a) and (b) 9 9 ) Thus, and.. ) 7. 7 ) 7 9.. ) 7 7 ) 7 7 7
Chapter Polnomial Functions. ). ) 7. ) 9 9 9. 7 7. 9 9 9 9. 7 7. 7. 9..
Section. Polnomial and Snthetic Division 7.. 7 7. f, k 9. f f f, k f f. f, k f f. f, k f f. f 7. h (a) (a) 9 9 9 9 97 f h 97 (b) f f (d) f 9 9 9 (b) (d) h h 7 h 99 7 99
Chapter Polnomial Functions 9. 7 7 Zeros:,,. 7 Zeros:,, 7 7.. Zeros: ±, Zeros:, ± 7. f ; Factors: (a) Both are factors of, f since the remainders are zero. (b) The remaining factor of f is. (d) Zeros: (e) f,, 7 9. f ; Factors: (a), Both are factors of f since the remainders are zero. (b) The remaining factors are and. f (d) Zeros: (e),,,
Section. Polnomial and Snthetic Division 9. f 9 ; Factors:, (a) 9 9 (b) Both are factors since the remainders are zero. 7 This shows that f so 7. The remaining factor is 7. f 7 (d) Zeros: 7,, f 7,. f ; Factors:, (a) Both are factors since the remainders are zero. (b) This shows that f so. The remaining factor is. (d) Zeros:,, f, f (e) (e) 9. f 7. (a) The zeros of f are and (b) An eact zero is. f ±.. ht t t 7t (a) The zeros of h are t, t.7, t.. (b) An eact zero is t. 7 ht t t t B the Quadratic Formula, the zeros of t t are ±. Thus, ht t t t t t t. 9. Thus,,. 7.,,
9 Chapter Polnomial Functions 7. (a) and (b) M.t.t 7.t Year, t Militar Personnel M 7 7 7 7 7 9 7 9 9 9 The model is a good fit to the actual data. (d).....7 M thousand 7.....77 No, this model should not be used to predict the number of militar personnel in the future. It predicts an increase in militar personnel until and then it decreases and will approach negative infinit quickl. 7. False. If 7 is a factor of f, then 7 is a zero of f. 77. True. The degree of the numerator is greater than the degree of the denominator. 79. n n 9. A divisor divides evenl into a dividend if the remainder n ) n 9 n 7 n 7 is zero. n n n 7 n n n n 9 n 7 n 7 n 9 n 7 9 n 7 n n 9. 9 To divide evenl, equal. c c c must equal zero. Thus, c must. f The remainder when k is zero since is a factor of f.
Section. Zeros of Polnomial Functions 9 7. 9 9. 7 7 7 9. 9. b ± b ac a ± 7 7 ± f 9. Note: An nonzero scalar multiple of f would also have these zeros. ± f 7 Note: An nonzero scalar multiple of f would also have these zeros. Section. Zeros of Polnomial Functions You should know that if f is a polnomial of degree n >, then f has at least one zero in the comple number sstem. You should know the Linear Factorization Theorem. You should know the Rational Zero Test. factors of constant term You should know shortcuts for the Rational Zero Test. Possible rational zeros factors of leading coefficient (a) Use a graphing or programmable calculator. (b) Sketch a graph. After finding a root, use snthetic division to reduce the degree of the polnomial. You should know that if a bi is a comple zero of a polnomial f, with real coefficients, then a bi is also a comple zero of f. You should know the difference between a factor that is irreducible over the rationals (such as 7) and a factor that is irreducible over the reals (such as 9). You should know Descartes s Rule of Signs. (For a polnomial with real coefficients and a non-zero constant term.) (a) The number of positive real zeros of f is either equal to the number of variations of sign of f or is less than that number b an even integer. (b) The number of negative real zeros of f is either equal to the number of variations in sign of f or is less than that number b an even integer. When there is onl one variation in sign, there is eactl one positive (or negative) real zero. You should be able to observe the last row obtained from snthetic division in order to determine upper or lower bounds. (a) If the test value is positive and all of the entries in the last row are positive or zero, then the test value is an upper bound. (b) If the test value is negative and the entries in the last row alternate from positive to negative, then the test value is a lower bound. (Zero entries count as positive or negative.)
9 Chapter Polnomial Functions Vocabular Check. Fundamental Theorem of Algebra. Linear Factorization Theme. Rational Zero. conjugate. irreducible; reals. Descarte s Rule of Signs 7. lower; upper. f. g. f i i The zeros are:,. The zeros are:,. The three zeros are:, i, i. 7. f 9. Possible rational zeros: ±, ± Zeros shown on graph:,, f 7 9 Possible rational zeros: Zeros shown on graph:,,, ±, ±, ±, ±9, ±, ±, ±, ±, ±, ± 9, ±, ±. f Possible rational zeros: ±, ±, ±, ± Thus, the rational zeros are,, and.. g Thus, the rational zeros of g are and ±.. ht t t t 7. Possible rational zeros: ±, ±, ±, ± t t t t t t t t t t t Thus, the rational zeros are and. C Possible rational zeros: ±, ± Thus, the rational zeros are and.
Section. Zeros of Polnomial Functions 9 9. f 9 9. z z z Possible rational zeros: ±, ±, ±, ±, ±, ±, ±, ±, Possible rational zeros: ±, ±, ± 9 9 9 9 9 7 7 7 9 9 ±, ±, ±, ±, ± 9, ± 9, ± 9, ± 9 9 z z z z z z The onl real zeros are and. Thus, the rational zeros are,, and ±.. 7 Possible rational zeros: ±, ±, ±, ±, ±, ± 7 9 9 7 7 9 7 7 The onl real zeros are,, and.. f 7. f (a) Possible rational zeros: ±, ±, ± (a) Possible rational zeros: ±, ±, ±, ±, ±, ± (b) (b) The zeros are:,, The zeros are:,, 9. f. f 7 (a) Possible rational zeros: ±, ±, ±, ±, ± (a) Possible rational zeros: ±, ±, ±, ±, ±, ±, ± (b), ±, ±, ±, ±, ± (b) The zeros are:,,, The zeros are:,,
9 Chapter Polnomial Functions. f. (a) From the calculator we have ± and ±.. (b) An eact zero is. f h 7 (a) h 7 From the calculator we have,, and ±.. (b) An eact zero is. 7 h 7. f i i 9. Note: f a, where a is an nonzero real number, has the zeros and ±i. f i i i i i 9 7 Note: f a 7, where a is an nonzero real number, has the zeros, and ± i.. If i is a zero, so is its conjugate, i.. f i i i i i 9 7 Note: f a 7, where a is an nonzero real number, has the zeros,, and ± i. f 7 (a) f 9 (b) f 9 f i i. f ) 7 f (a) f (b) f f i i Note: Use the Quadratic Formula for (b) and.
Section. Zeros of Polnomial Functions 9 7. f 7 Since i is a zero, so is i. i i i i i i i i i i The zero of is. The zeros of f are and ±i. 7 7 Alternate Solution Since ±i are zeros of f, i i is a factor of f. B long division we have: ) 7 7 7 7 Thus, f and the zeros of f are ±i and. 9. f 7 Since i is a zero, so is i. i i i i i i 7 i i i i i i i i The zeros of are and. The zeros of f are ±i,, and. Alternate Solution Since ±i are zeros of f, i i is a factor of f. B long division we have: ) 7 Thus, 7 f f i i and the zeros of f are ±i,, and.. g Since i is a zero, so is i. i i i i i i 7 i i i i The zero of is The zeros of g are ± i and.. Alternate Solution Since ± i are zeros of g, i i i i i is a factor of g. B long division we have: ) Thus, g and the zeros of g are ± i and.
9 Chapter Polnomial Functions. f Since i is a zero, so is i, and i i i i i is a factor of f. B long division, we have: ) Thus, f and the zeros of f are ± i,, and.. f 7. h i i B the Quadratic Formula, the zeros of h are The zeros of f are ±i. ± ±. h 9. f. fz z z 9 9 B the Quadratic Formula, the zeros of f z are i i The zeros of f are ± and ±i. z ± ± i. fz z iz i z iz i
Section. Zeros of Polnomial Functions 97. g. Possible rational zeros: B the Quadratic Formula, the zeros of are ± ±, ±, ±, ± ± i. The zeros of g are and ± i. g i i i i h Possible rational zeros: B the Quadratic Formula, the zeros of are ± ±, ±, ±, ± ± i. The zeros of h are and ± i. h i i i i 7. f 9 9. Possible rational zeros: ±, ±, ±, ±, ±, ±, ±, ± B the Quadratic Formula, the zeros of are 9 ± The zeros of f are ± i. and ± i. f i i i i g Possible rational zeros: ±, ±, ±, ±, ± g i i The zeros of g are and ±i. 7. f 9 7. f 7 9 i i i i The zeros of f are ±i and ±i. Possible rational zeros: ±, ±, ±, ±, ±, ±, ±7, ±7, ±, ±, ±7, ±7 Based on the graph, tr. B the Quadratic Formula, the zeros of 7 are ± 9 9 7 7 7 7 ± i. The zeros of f are and 7 ± i.
9 Chapter Polnomial Functions 7. f 77. Possible rational zeros: ±, ±, ±, ±, ± ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±, ±,, ± f Possible rational zeros: ±, ±, ± Based on the graph, tr. B the Quadratic Formula, the zeros of are ± ± i. The zeros of f are and ± i. Based on the graph, tr and. The zeros of are ±i. The zeros of f are,, and ±i. 79. g. Let f. Sign variations:, positive zeros: f Sign variations:, negative zeros: h Sign variations:, positive zeros: h Sign variations:, negative zeros:. g. f Sign variations:, positive zeros: g Sign variations:, positive zeros: or f Sign variations:, negative zeros: Sign variations:, negative zeros: 7. f 9. f (a) (a) 9 is an upper bound. (b) is a lower bound. is an upper bound. (b) 7 is a lower bound. 7
Section. Zeros of Polnomial Functions 99 9. f 9. Possible rational zeros: ±, ±, ± Thus, the zeros are and. f Possible rational zeros: ±, ±, ±, ±, ±, ±, ±, ± Thus, the onl real zero is. 9. P 9 97. 9 The rational zeros are and ±. ± f The rational zeros are and ±. 99. f. Rational zeros: Irrational zeros: Matches (d). f Rational zeros: Irrational zeros: Matches (b)., ±. (a) 9 Volume of bo 7 V 9 Length of sides of squares removed The volume is maimum when.. The dimensions are: length.. width 9.. height.. cm. cm. cm (b) (d) V l w h 9 9 Since length, width, and height must be positive, we have < < 9 for the domain. 9, 7 The zeros of this polnomial are, and. cannot equal since it is not in the domain of V. [The length cannot equal and the width cannot equal 7. The product of 7 so it showed up as an etraneous solution.] Thus, the volume is cubic centimeters when centimeter or 7 centimeters.. P 7,,,, 7, 7,, The zeros of this equation are.,., and.. Since, we disregard.. The smaller remaining solution is.. The advertising epense is $,.
Chapter Polnomial Functions 7. (a) Current bin: V cubic feet 9. (b) New bin: V cubic feet V 9 9 9 The onl real zero of this polnomial is. All the dimensions should be increased b feet, so the new bin will have dimensions of feet b feet b feet. C, C is minimum when. The onl real zero is or units.. 9,,., Thus,. 9,,. P R C p C.,., ±., ±,i Since the solutions are both comple, it is not possible to determine a price p that would ield a profit of 9 million dollars.. False. The most nonreal comple zeros it can have is two and the Linear Factorization Theorem guarantees that there are linear factors, so one zero must be real.. g f. This function would have the same zeros 7. g f. The graph of g is a horizontal shift as f so r, r, and r are also zeros of g. of the graph of f five units to the right so the zeros of g are r, r, and r. 9. g f. Since g is a vertical shift of the graph of f, the zeros of g cannot be determined.. f k ± k ± ± k ± k ± k (a) For there to be four distinct real roots, both k and ± k must be positive. This occurs when < k <. Thus, some possible k-values are k, k, k, k, k, etc. (b) For there to be two real roots, each of multiplicit, k must equal zero. Thus, k. For there to be two real zeros and two comple zeros, k must be positive and k must be negative. This occurs when k <. Thus, some possible k-values are k, k, k, etc. (d) For there to be four comple zeros, ± k must be nonreal. This occurs when k >. Some possible k-values are k, k, k 7., etc.
Section. Zeros of Polnomial Functions. Zeros: f (, ),, (, ( (, ) An nonzero scalar multiple of f would have the same three zeros. Let g af, a >. There are infinitel man possible functions for f.. Answers will var. Some of the factoring techniques are:. Factor out the greatest common factor.. Use special product formulas. a b a ba b a ab b a b a ab b a b a b a ba ab b a b a ba ab b. Factor b grouping, if possible.. Factor general trinomials with binomial factors b guess-and-test or b the grouping method.. Use the Rational Zero Test together with snthetic division to factor a polnomial.. Use Descartes s Rule of Signs to determine the number of real zeros. Then find an zeros and use them to factor the polnomial. 7. Find an upper and lower bounds for the real zeros to eliminate some of the possible rational zeros. Then test the remaining candidates b snthetic division and use an zeros to factor the polnomial. 7. (a) (b) f bi bi b f a bi a bi a bi a bi a bi a a b 9. i i i i 9i. i 7i i i i i. g f. g f 7. g f (, ) (, ) (, ) (, ) Horizontal shift two units to the right (, ) (, ) (, ) (, ) Vertical stretch (each -value is multiplied b ) (, ) (, ) (, ) (, ) Horizontal shrink each -value is multiplied b
Chapter Polnomial Functions Section. Mathematical Modeling and Variation You should know the following the following terms and formulas. Direct variation (varies directl, directl proportional) (a) k (b) k n as nth power Inverse variation (varies inversel, inversel proportional) (a) k (b) k n as nth power Joint variation (varies jointl, jointl proportional) (a) z k (b) z k n m as nth power of and mth power of k is called the constant of proportionalit. Least Squares Regression Line a b. Use our calculator or computer to enter the data points and to find the best-fitting linear model. Vocabular Check. variation; regression. sum of square differences. correlation coefficient. directl proportional. constant of variation. directl proportional 7. inverse. combined 9. jointl proportional. 77.t,9 Year Actual Number Model (in thousands) (in thousands) 99, 7, 99 9, 9,7 99,,9 Number of emploees (in thousands),,,,, Year ( 99) t 99,,7 99,9, The model is a good fit for the actual data. 997,97, 99 7,7, 999 9, 9,9,,,7,,,. Using the points, and,, we have.. Using the points, and,, we have.
Section. Mathematical Modeling and Variation 7. (a) and (b) Length (in feet) t Year ( 9).t.7 (d) The models are similar. (e) When t, we have: Model in part (b): feet Model in part :. feet (f) Answers will var. t 9. (a) and The model is a good fit to the actual data. r.9 (b) S.t (d) For, use t : S $. million For 7, use t 7: S $77. million (e) Each ear the annual gross ticket sales for Broadwa shows in New York Cit increase b approimatel $. million.. The graph appears to represent, so varies inversel as.. k. k k k 7. k 9. k k k
Chapter Polnomial Functions. The table represents the equation.. k. 7 k 7 k 7 This equation checks with the other points given in the table. k k k 7. k 9. I kp. k k 7. k k k. k I.P k When inches,. centimeters. When inches,. centimeters.. k. k,. k.., $7 The propert ta is $7. (a) (b) d kf 7.. k k d F d 9. meter. F F F 7 newtons d kf.9 k k.7 d.7f When the distance compressed is inches, we have.7f F 9.7. No child over 9.7 pounds should use the to. 9. A kr. k. F kg r. P k V 7. F km m r 9. A bh. The area of a triangle is jointl proportional to its base and height. V r The volume of a sphere varies directl as the cube of its radius.. r d t. A kr 7. k Average speed is directl proportional to the distance and inversel proportional to the time. 9 k k A r 7 k k
Section. Mathematical Modeling and Variation 9. F krs. k k F rs z k k k k z. d kv. k k. d.v..v v.. v. mihr..7 k....7 l r kl A, A d r r kl d k.7 r.7 l...7 l. l feet 7. W kmh. k. k. 9.. W 9.mh When m kilograms and h. meters, we have W 9.. 7 joules. 9. v k A v k.7a A k The velocit is increased b one-third. 7. (a) C Temperature (in C) Depth (in meters) d (b) Yes, the data appears to be modeled (approimatel) b the inverse proportion model.. k.9 k. k. k.9 k k k k k k CONTINUED
Chapter Polnomial Functions 7. CONTINUED Mean: (d) k, (e) Model: d d C d meters 7..7. (a). 7. False. will increase if k is positive and will decrease if k is negative. 77. False. The closer the value of r is to, the better the fit. (b).7..7 microwatts per sq.cm. 79. The accurac of the model in predicting prize winnings is questionable because the model is based on limited data.. > 7 > 7 9 >. < 9. 9 < < 9 < < < < f (a) (b) f f f 7 Review Eercises for Chapter. (a) (b) Vertical stretch Vertical stretch and a reflection in the -ais CONTINUED
Review Eercises for Chapter 7. CONTINUED Vertical shift two units upward (a) (d) Horizontal shift two units to the left. g. f Verte:, 7 Verte:, Ais of smmetr: -intercepts:,,, Ais of smmetr: ± ± -intercepts: ±, 7. ft t t 9. h t t t t Verte:, Ais of smmetr: t t t t Verte:, t ± Ais of smmetr: t ± t-intercepts: ±, No real zeros -intercepts: none
Chapter Polnomial Functions. h. Verte:, Ais of smmetr: B the Quadratic Formula, -intercepts: ±, ±. f Verte:, Ais of smmetr: B the Quadratic Formula, -intercepts: ±, ±.. Verte:, f a Point:, a a a Thus, f. 7. Verte:, f a Point:, a a Thus, f. 9. (a) A since (d) (b) Since the figure is in the first quadrant and and must be positive, the domain of. A Area is < <. 7 (e) 9 The maimum area of occurs at the verte when and. A The maimum area of occurs when and.
Review Eercises for Chapter 9. R p p. C 7,. (a) R $, The minimum cost occurs at the verte of the parabola. R $,7 R $, (b) The maimum revenue occurs at the verte of the parabola. b $ a R $, The revenue is maimum when the price is $ per unit. The maimum revenue is $,. Verte: b 9 units a. Approimatel 9 units should be produced each da to ield a minimum cost.., f 7., f 9., f 7 7 Transformation: Reflection in the -ais and a horizontal shift four units to the right Transformation: Reflection in the -ais and a vertical shift two units upward Transformation: Horizontal shift three units to the right. f 9. The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. g The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right.. f 7. f t t t 9 7 9 t t tt Zeros:, 7, all of multiplicit (odd multiplicit) Turning points: Zeros: t, ± all of multiplicit (odd multiplicit) Turning points: 9. f Zeros: of multiplicit (even multiplicit) of multiplicit (odd multiplicit) Turning points:
Chapter Polnomial Functions. f. f (a) The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. (a) The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right. (b) Zero: (b) Zeros:,, f f 7 (d) (d) (, ) (, ) (, ) (, ). f 7. f f 7 7 f 9 The zero is in the interval,. Zero:.9 There are two zeros, one in the interval, and one in the interval, Zeros:.,.77 9. ) Thus,.. Thus, ).. ) Thus,.. Thus, 7 7.
Review Eercises for Chapter 7. 9 Thus, 9. 9. f 9 (a) 9 (b) 9 Yes, is a zero of f. Yes, is a zero of f. 9 (d) 9 9 9 9 Yes, is a zero of f. No, is not a zero of f.. f (a) (b) 9 7 9 9 Thus, f. f 9. f ; Factor:. (a) 7 Yes, is a factor of f. (b) 7 7 The remaining factors of f are 7 and. (d) Zeros: 7,, (e) f 7 f 7 Factors: (a) (b) Both are factors since the remainders are zero. The remaining factors are and. (d) Zeros: (e), f,,, 7 9
Chapter Polnomial Functions 7. f 9. Zeros:, f 9 Zeros:, 7. f i i 7. Zeros:,, i, i f Possible rational zeros: ±, ±, ±, ±, ± ±, ±, ±, ±, ±, ±, ±, 7. f 77. f 7 Possible rational zeros: ±, ±, ±, ±, ±9, ± Possible rational zeros: ±, ±, ±, ± The zeros of f are,, and. 9 7 9 7 9 The zeros of f are and. 79. f Possible rational zeros: ±, ±, ±, ±, ±, ± The real zeros of f are, and. f i i. Multipl b to clear the fraction. Since i is a zero, so is i. 7 Note: f a 7, where a is an real nonzero number, has zeros,, and ±i.. f, Zero: i Since i is a zero, so is i. i i i i i i i i i i f i i, Zeros: ± i,
Review Eercises for Chapter. g 7, Zero: i 7. Since i is a zero, so is i i i i i i i i i i i 7 i i i i f Zeros:,, g i i i i Zeros: ± i,, 9. g, Zero: 9. g B the Quadratic Formula the zeros of are ± i. The zeros of g are of multiplicit, and ± i. g i i i i f (a) 7 (b) The graph has two -intercepts, so there are two real zeros. The zeros are and.. 9. h 9. g 9 (a) g has two variations in sign, so g has either two or no positive real zeros. (b) The graph has one -intercept, so there is one real zero.. g 9 g has one variation in sign, so g has one negative real zero. 97. f (a) Since the last row has all positive entries, is an upper bound. (b) 7 Since the last row entries alternate in sign, is a lower bound.
Chapter Polnomial Functions 99. I.9t 7.. (a) Median income (in thousands of dollars) 7 9 Year ( 99) (b) The model is a good fit to the actual data. r.99 t D km. F ks k. k D m In miles, D. kilometers. In miles, D kilometers. If the speed is doubled, F ks F ks. The force will be changed b a factor of.. T k r k k 9 T 9 r When t mph, T 9.7 hours hours, minutes. 7. False. A fourth-degree polnomial can have at most four zeros and comple zeros occur in conjugate pairs. 9. The maimum (or minimum) value of a quadratic function is located at its graph s verte. To find the verte, either write the equation in standard form or use the formula b a, f b a. If the leading coefficient is positive, the verte is a minimum. If the leading coefficient is negative, the verte is a maimum. Problem Solving for Chapter. (a) (i) g (ii) g Zeros:, Zeros:, (iii) g (iv) g (v) Zeros:, g B the Quadratic Formula, ± 7. Zeros: ± 7 (vi) Zeros: g ± 7i B the Quadratic Formula,. Zeros:, ± 7i CONTINUED
Problem Solving for Chapter. CONTINUED (b) (i) f (ii) f (iii) f (iv) f (v) f (vi) f 9 9 is an -intercept of f in all si graphs. All the graphs, ecept (iii), cross the -ais at. (i) other -intercepts: (ii) other -intercepts:,,,,,, (iii) other -intercepts:, (iv) other -intercepts: No additional -intercepts (v) other -intercepts:.,,., (vi) other -intercepts: No additional -intercepts (d) The -intercepts of f g are the same as the zeros of g plus.. f a b c d a ak b ak bk c k) a b c d a ak ak b c ak b ak bk ak bk c d ak bk c ak bk ck ak bk ck d Thus, f a b c d ka ak b ak b c ak bk ck d and f k ak bk ck d. Since the remainder r ak bk ck d, f k r.
Chapter Polnomial Functions. V l w h Possible rational zeros: ±, ±, ±, ±, ±, ± or ± i + Choosing the real positive value for we have: and. The dimensions of the mold are inches inches inches. 7. False. Since f dq r, we have f r q d d. The statement should be corrected to read f since f f q. 9. (a) Slope 9 Slope of tangent line is less than. (b) Slope Slope of tangent line is greater than. Slope Slope of tangent line is less than.. (d) Slope (e).. h h h h h h, h Slope h, h... f h f h The results are the same as in (a). (f) Letting h get closer and closer to, the slope approaches. Hence, the slope at, is.
Problem Solving for Chapter 7. Let length of the wire used to form the square. Then length of wire used to form the circle. (a) Let s the side of the square. Then s s and the area of the square is s. Let r the radius of the circle. Then and the area of the circle is r r r. The combined area is: (b) Domain: Since the wire is cm,. A A,, The minimum occurs at the verte when cm and A cm. The maimum occurs at one of the endpoints of the domain. When, A 79 cm. When, A cm. Thus, the area is maimum when cm. (d) Answers will var. Graph A to see where the minimum and maimum values occur.
Chapter Polnomial Functions Practice Test for Chapter. Sketch the graph of f and identif the verte and the intercepts.. Find the number of units that produce a minimum cost C if C. 9,.. Find the quadratic function that has a maimum at, 7 and passes through the point,.. Find two quadratic functions that have -intercepts, and,.. Use the leading coefficient test to determine the right and left end behavior of the graph of the polnomial function f 7.. Find all the real zeros of f. 7. Find a polnomial function with,, and as zeros.. Sketch f. 9. Divide 7 b using long division.. Divide b.. Use snthetic division to divide b.. Use snthetic division to find f given f 7.. Find the real zeros of f 9.. Find the real zeros of f 9 9.. List all possible rational zeros of the function f.. Find the rational zeros of the polnomial f 9. 7. Write f as a product of linear factors.. Find a polnomial with real coefficients that has, i, and i as zeros. 9. Use snthetic division to show that i is a zero of f 9.. Find a mathematical model for the statement, z varies directl as the square of and inversel as the square root of.