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Math 150, Fall 2008, Benjamin
Aurispa
Chapter 3: Polynomial and Rational Functions
3.1 Polynomial Functions and Their Graphs
A polynomial function of degree n is a function of the form
P (x) = an xn + an−1 xn−1 + · · · + a1 x + a0
The numbers a0 , a1 , . . . , an are called the coefficients of the polynomial.
The number a0 is called the constant coefficient (or constant term).
The number an , the coefficient of the highest power of x, is called the leading coefficient and the term
an xn is called the leading term.
The end behavior of a polynomial is what happens to the graph as x becomes very large in either the
positive or negative direction. In other words, what happens as x → ∞ and as x → −∞.
x → ∞ is read “as x approaches infinity”
x → −∞ is read “as x approaches negative infinity”
The end behavior of a general polynomial is determined by
(1) the degree of the polynomial and
(2) the sign of the leading coefficient.
This gives us four cases:
1. Even degree, positive leading coefficient. (Think of x2 .)
As x → ∞,
and as x → −∞,
.
2. Even degree, negative leading coefficient. (Think of −x2 .)
As x → ∞,
and as x → −∞,
.
3. Odd degree, positive leading coefficient. (Think of x3 .)
As x → ∞,
and as x → −∞,
.
4. Odd degree, negative leading coefficient. (Think of −x3 .)
and as x → −∞,
As x → ∞,
.
1
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Math 150, Fall 2008, Benjamin
Aurispa
Determine the end behavior of the following polynomials:
• P (x) = −2x10 − 20000x + 30000
• P (x) = −x5 + 5x3 − 4x
• P (x) = (3x + 2)2 (x − 1)3 (x + 4)
We call a number c a zero of a polynomial P (x) if P (c) = 0.
In other words, the zeros of a polynomial are the x-intercepts.
Example: Find the zeros of P (x) = x2 + 9x + 20?
The following 4 statements all mean the same thing for a polynomial P .
1. c is a zero of P .
2. x = c is a solution to the equation P (x) = 0.
3. x − c is a factor of P .
4. x = c is an x-intercept of the graph of P .
Intermediate Value Theorem for Polynomials:
If P is a polynomial function and P (a) and P (b) have opposite signs, then there exists at least one value c
between a and b for which P (c) = 0, ie, there exists a value of c between a and b where c is a zero of P .
What this tells us is that if I have two consecutive zeros, then the graph must lie entirely above or entirely
below the x-axis between those zeros.
2
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Math 150, Fall 2008, Benjamin
Aurispa
We call c a zero of multiplicity m when the factor x − c occurs exactly m times in the factorization of P .
What are the multiplicities of the zeros of P (x) = (x − 1)2 (x + 2)3 ?
If c is a zero of multiplicity m, there are 2 cases:
1. If m is odd, then the graph of P actually crosses the x-axis at this zero.
2. If m is even, then the graph of P does not completely cross the x-axis at this zero. It only touches it.
To sketch the graph of a polynomial:
1. Factor the polynomial and determine the zeros (x-intercepts) of the graph.
2. Plot the x and y-intercepts. You may also want to plot some other points.
3. Determine the end behavior of the polynomial.
4. Sketch the graph that passes through the intercepts and any other points, making sure that the
polynomial has the correct end behavior.
Sketch the graph of P (x) = −x3 − 2x2 + 8x.
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3
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Math 150, Fall 2008, Benjamin
Aurispa
Sketch the graph of P (x) = 2x4 − 20x2 + 18.
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Sketch the graph of P (x) = (x − 1)2 (x + 2)3 .
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4
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Math 150, Fall 2008, Benjamin
Aurispa
3.2 Dividing Polynomials
Long Division of Polynomials
Dividing polynomials by long division is very similar to doing long division with numbers. There is still a
dividend, divisor, quotient, and remainder. With polynomials, we always look first at the leading terms of
the polynomials we’re dividing.
Examples
• Divide 6x3 − 20x2 + 31x − 17 by 3x − 4.
We can write the answer in the form P (x) = D(x)Q(x) + R(x), where D(x) is the divisor, Q(x) is the
quotient, and R(x) is the remainder. (R(x) must have degree less than D(x).)
Another way to express this is:
P (x)
R(x)
= Q(x) +
D(x)
D(x)
• Let P (x) = x5 + x4 − 2x3 + x + 1 and D(x) = x2 + x − 1. Find Q(x) and R(x).
5
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Math 150, Fall 2008, Benjamin
Aurispa
Synthetic division is a quick way to divide polynomials where the divisor is of the form x − c. It allows us
to just work with the coefficients instead of keeping track of the variables. Remember to include zeros for
the powers of x that are not present.
Examples. Use synthetic division to find the quotient and remainder
3x3 − 12x2 − 9x + 1
x−5
x4 − 16
x+2
Factor Theorem: c is a zero of P if and only if x − c is a factor of P (x).
Factor the polynomial P (x) = 3x4 − x3 − 21x2 − 11x + 6 completely if it is known that
of the polynomial.
Find a polynomial of degree 4 that has zeros: −2, 0, 2, 4.
Find a polynomial of degree 3 with integer coefficients that has zeros:
6
1
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2, −3,
and 1.
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and −2 are zeros