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P. 251-25612-16, 40-4412) in (-, -1], de [-1, ) 13) in (- , 2], [1, ), de [-2, 1]14) in (- , -2] , [0, 1], de [-2, 0], [1, ]15) in [-8, 0], [8, ), de (- , 8], [0, 8]16) in (- , 0], de [0, )40) lmx: 0, lmn: -5, no amx, amn: 541) lmx: 17, 27,lmn: -10, 2442) lmx: 2, amx: 243) lmn: -3.2, amn:-3.2, amx: 344) lmx: 3.1, lmn: -3.1, amx: 3.1, amn: -3.1

1P. 251-25645-48, 70-90 even45) lmx: 1, lmn: -1, amx: 1, amn: -146) lmx: 0.5, 2, lmn:-0.5, -2, amx: 0.5, 2, amn: -0.5, -247) lmn: -2, lmx: 1, 2, amn: -2, amx: 248) lmx: 2.5, lmn: 2, -12Amn: -1270)even72) odd74) Neither76) Even78) even80) Odd82) neither84) Neither86) Even88) Odd90) neither 2P. 268- 273 #9-18, 67-739) a. 2 turning points, x-intercept = -3b. Positivec. Minimum degree of 310)4 turning points, x-intercepts = -2, -1, 0, 1, 2NegativeMinimum degree is 5

311-1211)One turning point, x-intercepts = -1, 2Positive 212)No turning points, x-intercept = Negative1413-1513) (a) d (b) (1,0) (c) x=1 (d) (1,0) (e) (1,0)14) (a) c (b) (-1, -2), (1,2)(c) x= -1.8, 0, 1.8(d) (-1, -2), (1,2) (e) none15) (a) b(b) (-3, 27), (1,-5)(c) x=0, 1.9(d) (-3, 27), (1,-5) (e) none516-1816) (a) f(b) (-2, -16), (0, 0), (2, -16)(c) x= -2.8, 0, 2.8 (d) (-2, -16), (0, 0), (2, -16) (e) (-2, -16), (2, -16)17) (a) a(b) (-2, 16), (0,0), (2, 16)(c) x= -2.8, 0, 2.8 (d) (-2, 16), (0,0), (2, 16)(e) (-2, 16), (2, 16)18) (a) e(b) (-2, 1), (-1, -2), (0,0), (1, -3)(c) x= -2.2, 0, 1.2(d) (-2, 1), (-1, -2), (0,0), (1, -3) (e) none667-7367) f(-2)=5, f(1)=068) f(-1)=0, f(0)= -.7, f(3)=269) f(-1)=-1, f(1) =1, f(2)= -270) f(-2)=0, f(0)=-3, f(2)=271) f(-3)=-63, f(1)=3, f(4)=1072) f(-4)=16, f(0)=2, f(4)=-1273) f(-2)=6, f(1)=7, f(2)=974.3: Real Zeroes of Polynomials Functions8Remember

9Divide polynomialsDivide by a monomial(3x3+6x2+7)/(2x)

10Divide polynomialsDividing by binomials or higher(2x3+4x2-x+6)/(x+1)

11Things to rememberIf f(x) has a degree of 1 or greater, thenThe graph of y=f(x) has a x-intercept k.A zero of f(x) is k. Basically, f(k)=0.A factor of f(x) is (x-k).13MultiplicityEven multiplicities (squared, to the fourth, etc.) intersect but do not cross at zero. The turn back.Odd multiplicities (1st, 3rd, etc.) cross at zero.

Total multiplicity must add up to the degree.14Multiplicity of 1

16Even multiplicity

17Odd multiplicity (greater than 1)

18Find zeros on this graph.

19Check this outWhat makes f(x) equal zero on this graph.

20Things to rememberIf f(x) has a degree of 1 or greater, thenThe graph of y=f(x) has a x-intercept k.A zero of f(x) is k. Basically, f(k)=0.A factor of f(x) is (x-k).21Factoring2x3-4x2-10x+12, given that k=2 and is a zero.224.4: The Fundamental Theorem of Algebra23Complex numbersImaginary numbers: i = (-1), i2 = -1Complex numbers: all real number and all imaginary numbersAdd and subtract as if i is a variable2i-1+3i+2

24Multiplying numbers(4-i)(5+i)

25Dividing complex numbersDivide: eliminate by multiplying the numerator and the denominator by the conjugate of the denominator (change the middle sign of a + bi)2-i/(3+i)26Solving quadratics with imaginary solutionsX2+2x+10

x = -2 (22-4(1)(10) 2= -2 (4-40)=-2 -36 2 2= -2 6i 2= -1 3i

27Fundamental Theorem of AlgebraA polynomial function f(x) of degree n 1 has at least one complex (remember real or imaginary) zeroA polynomial of degree n has at most n distinct zeros

28Conjugate zeros theoremIf a polynomial has only real coefficients and if a+bi is a zero of f(x), then a-bi is also a zero of f(x)29Your assignmentPage 288-29215-1931-3847-5063-65

P. 302-305#10-36 even#48, 54, 60#70-76 even30