Download Math 180 Chapter 3 Lecture Notes

Math 180
Chapter 3
Lecture Notes
Professor Miguel Ornelas
1
M. Ornelas
Math 180 Lecture Notes
Section 3.1
Section 3.1
Polynomial Functions of Degree Greater Than 2
Degree of f
Form of f (x)
Graph of f
0
1
2
f (x) = a0
f (x) = a1 x + a0
f (x) = a2 x2 + a1 x + a0
A horizontal line
A line with slope a1
A parabola with a vertical axis
Let f (x) = 2x3 + 3.
y
Sketch the graph of f
x
f (x)
x
Let f (x) = −2x3 + 3.
y
Sketch the graph of f
x
f (x)
x
Intermediate Value Theorem for Polynomial Functions
If f is a polynomial function and f (a) , f (b) for a < b, then f takes on every value between f (a) and f (b) in the
interval [a, b].
Show that f (x) = −x4 + 3x3 − 2x + 1 has a zero between 2 and 3.
Section 3.1 continued on next page. . .
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M. Ornelas
Math 180 Lecture Notes
Section 3.1 (continued)
Let f (x) = x3 + x2 − 5x − 5. Find all values of x such that f (x) > 0 and all x such that f (x) < 0, and then sketch the
graph of f .
y
Zeros
x
Let f (x) = x4 − 6x2 + 8. Find all values of x such that f (x) > 0 and all x such that f (x) < 0, and then sketch the graph
of f .
y
Zeros
x
3
M. Ornelas
Math 180 Lecture Notes
Section 3.2
Properties of Division
Division Algorithm for Polynomials
If f (x) and p(x) are polynomials, then there exist unique polynomials q(x) and r(x) such that
f (x) = p(x) · q(x) + r(x),
where the polynomial q(x) is the quotient , and r(x) is the remainder in the division of f (x) and p(x).
Find the quotient and remainder if f (x) = 3x4 + 2x3 − x2 − x − 6 is divided by p(x) = x2 + 1
Remainder Theorem
If a polynomial f (x) is divided by x − c, then the remainder is f (c).
If f (x) = 2x3 + 4x2 − 3x − 1,use the remainder theorem to find f (3).
Factor Theorem
A polynomial f (x) has a factor x − c if and only if f (c) = 0.
Show that the x − 2 is a factor of f (x) = x3 + x2 − 11x + 10.
Section 3.2 continued on next page. . .
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Section 3.2
M. Ornelas
Math 180 Lecture Notes
Section 3.2 (continued)
Find a polynomial f (x) of degree 3 that has zeros -3, 0, and 4.
Use synthetic division to find the quotient q(x) and remainder r if the polynomial −2x4 + 10x − 3 is divided by x − 3.
If f (x) = 27x5 + 2x2 + 1, use synthetic division to find f
!
1
.
3
Use synthetic division to show that 3 is a zero of the polynomial f (x) = 4x3 − 9x2 − 8x − 3.
Section 3.2 continued on next page. . .
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M. Ornelas
Math 180 Lecture Notes
Section 3.2 (continued)
Section 3.3
Zeros of Polynomials
Fundamental Theorem of Algebra
If a polynomial f (x) has positive degree and complex coefficients, then f (x) has at least one complex zero.
Complete Factorization Theorem for Polynomials
If f (x) is a polynomial of degree n > 0, then there exist n complex numbers c1 , c2 , ..., cn such that
f (x) = a(x − c1 )(x − c2 ) · · · (x − cn ),
where a is the leading coefficient of f (x). Each number ck is a zero of f (x).
Theorem on the Maximum Number of Zeros of a Polynomial
A polynomial of degree n > 0 has at most n different complex zeros.
Find a polynomial f (x) in factored form that has degree 3; has zeros 2, -1, and 3; and satisfies f (1) = 5.
Find the zeros of the polynomial f (x) = 16x5 −40x4 +25x3 , state the multiplicity of each, and then sketch the graph of f .
y
Zeros
x
Section 3.3 continued on next page. . .
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M. Ornelas
Math 180 Lecture Notes
Section 3.3 (continued)
Show that -2 of multiplicity 3, is a zero of f (x) = x4 + 5x3 + 6x2 − 4x − 8, and express f (x) as a product of linear
factors.
Descartes’ Rule of Signs
Let f (x) be a polynomial with real coefficients and a nonzero constant term.
1. The number of positive real zeros of f (x) either is equal to the number of variations of sign in f (x) or is less
than that number by an even integer.
2. The number of negative real zeros of f (x) either is equal to the number of variations of sign in f (−x) or is less
than that number by an even integer.
Discuss that number of possible positive and negative real solutions and imaginary solutions of the equation
3x4 + 2x3 − 4x + 2 = 0
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M. Ornelas
Math 180 Lecture Notes
Section 3.4
Section 3.4
Complex and Rational Zeros of Polynomials
Theorem on Conjugate Pair Zeros of a Polynomial
If a polynomial f (x) of degree n > 1 has real coefficients and if z = a + bi is a complex zero, then the conjugate
z̄ = a − bi is also a zero of f (x).
Find a polynomial f (x) of degree 4 that has real coefficients and zeros 4 + 3i and −2 + i.
Theorem on Rational Zeros of a Polynomial
If the polynomial f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 has integer coefficients and if c/d is a rational zero of f (x)
such that c and d have no common prime factor, then
1. the numerator c of the zero is a factor of the constant term a0
2. the denominator d of the zero is a factor of the leading coefficient an
Show that the polynomial f (x) = 3x3 − 4x2 + 7x + 5 has no rational zeros.
Section 3.4 continued on next page. . .
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M. Ornelas
Math 180 Lecture Notes
Find all rational solutions of the equation x3 + x2 − 14x − 24 = 0.
Find all rational solutions of the equation 6x5 + 19x4 + x3 − 6x2 = 0.
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Section 3.4 (continued)
M. Ornelas
Math 180 Lecture Notes
Section 3.5
Rational Functions
A function f is a rational function if
f (x) =
g(x)
,
h(x)
where g(x) and h(x) are polynomials.
Use arrow notation to describe the end behavior of the function.
1.
f (x) =
2
x−3
2.
f (x) =
2x
x−3
Identify any vertical asymptotes, horizontal asymptotes, and holes.
1.
f (x) =
−2(x + 5)(x − 6)
(x − 3)(x − 6)
2.
f (x) =
Guidelines for Sketching the Graph of a Rational Function
g(x)
Assume that f (x) =
.
h(x)
1. Find the x-intercepts
2. Find the vertical asymptotes
3. Find the y-intercept
Section 3.5 continued on next page. . .
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2(x + 4)(x + 2)
5(x + 2)(x − 1)
Section 3.5
M. Ornelas
Math 180 Lecture Notes
4. Find the horizontal asymptotes
5. If there is a horizontal asymptote, determine whether it intersects the graph
6. Sketch the graph
Sketch the graph f (x) =
4x
2x − 5
y
x
Sketch the graph f (x) =
x2
x+1
+ 2x − 3
y
x
11
Section 3.5 (continued)