P. 251-25612-16, 40-44 12) 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: 5 41) lmx: 17, 27,lmn: -10, 24 42) lmx: 2, amx: 2 43) lmn: -3.2, amn:-3.2, amx: 3 44) lmx: 3.1, lmn: -3.1, amx: 3.1, amn: -3.1
P. 251-25645-48, 70-90 even 45) lmx: 1, lmn: -1, amx: 1, amn: -1 46) lmx: 0.5, 2, lmn:-0.5, -2, amx: 0.5, 2, amn: -0.5, -2 47) lmn: -2, lmx: 1, 2, amn: -2, amx: 2 48) lmx: 2.5, lmn: 2, -12 Amn: -12 70)even 72) odd 74) Neither 76) Even 78) even 80) Odd 82) neither 84) Neither 86) Even 88) Odd 90) neither
P. 268- 273 #9-18, 67-73 9) a. 2 turning points, x-intercept = -3 b. Positive c. Minimum degree of 3 10) • 4 turning points, x-intercepts = -2, -1, 0, 1, 2 • Negative • Minimum degree is 5
11-12 11) • One turning point, x-intercepts = -1, 2 • Positive • 2 12) • No turning points, x-intercept = ½ • Negative • 1
13-15 13) (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) none 15) (a) b (b) (-3, 27), (1,-5) (c) x=0, 1.9 (d) (-3, 27), (1,-5) (e) none
16-18 16) (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) none
67-73 67) f(-2)=5, f(1)=0 68) f(-1)=0, f(0)= -.7, f(3)=2 69) f(-1)=-1, f(1) =1, f(2)= -2 70) f(-2)=0, f(0)=-3, f(2)=2 71) f(-3)=-63, f(1)=3, f(4)=10 72) f(-4)=16, f(0)=2, f(4)=-12 73) f(-2)=6, f(1)=7, f(2)=9
4.3: Real Zeroes of Polynomials Functions
Remember…
Divide polynomials • Divide by a monomial (3x3+6x2+7)/(2x)
Divide polynomials • Dividing by binomials or higher (2x3+4x2-x+6)/(x+1)
Things to remember If f(x) has a degree of 1 or greater, then… • The 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).
Multiplicity Even 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.
Multiplicity of 1
Even multiplicity
Odd multiplicity (greater than 1)
Find zeros on this graph.
Check this out What makes f(x) equal zero on this graph.
Things to remember If f(x) has a degree of 1 or greater, then… • The 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).
Factoring… 2x3-4x2-10x+12, given that k=2 and is a zero.
4.4: The Fundamental Theorem of Algebra
Complex numbers • Imaginary numbers: i = √(-1), i2 = -1 • Complex numbers: all real number and all imaginary numbers • Add and subtract as if i is a variable 2i-1+3i+2
Multiplying numbers (4-i)(5+i)
Dividing complex numbers Divide: 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)
Solving quadratics with imaginary solutions X2+2x+10 x = -2 ± √(22-4(1)(10) 2 = -2 ± √(4-40) =-2 ± √-36 2 2 = -2 ± 6i 2 = -1 ± 3i
Fundamental Theorem of Algebra • A polynomial function f(x) of degree n≥ 1 has at least one complex (remember– real or imaginary) zero • A polynomial of degree n has at most n distinct zeros
Conjugate zeros theorem If 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)
Your assignment • Page 288-292 15-19 31-38 47-50 63-65 • P. 302-305 #10-36 even #48, 54, 60 #70-76 even
