Download Section 5.1: Polynomial Functions

Section 5.1: Polynomial Functions
• Def: A polynomial function is a function of the form:
f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0
where an , an−1 , . . . , a1 , a0 are real numbers and the exponents are all nonnegative integers. The domain of a polynomial function is the set of all real
numbers.
• Def: The degree of a polynomial function is the largest exponent of x that
appears in the function.
• A polynomial function is an example of what is called a continuous function.
A continuous function is essentially a function with no jumps, gaps, or holes
in its graph.
• Def: A power function of degree n is a function of the form
f (x) = axn
where a is a non-zero real number and n is a non-negative integer.
• The graph of a power function depends on whether the the exponent is even
or odd.
1. n is even
1
2. n odd
• ex. Graph:
(a) f (x) = (x − 2)5
2
(b) f (x) = x4 + 2
(c) f (x) = 3 − (x + 2)4
• Def: If y = f (x) is a polynomial function, then the x-intercepts of f are
sometimes called the real zeros of f or the roots of f .
• The zeros of f are the values of x for which f (x) = 0; i.e. if r is a real number
such that f (r) = 0, then r is a real zero of f . Conversely, if r is a zero of
3
f , then (r, 0) is an x-intercept of f . Furthermore, if r is a real zero of f ,
then(x − r) is a factor of f .
• ex. Form a polynomial whose zeros and degree are given below. Leave your
answer in factored form.
Zeros: −4, 0, 2; degree: 3
• Def: If the factor (x − r) occurs more than once in a function, then r is called
a repeated zero of f or a multiple zero of f . More precisely, if (x − r)m is a
factor of a polynomial function f and (x − r)m+1 is not a factor of f , then r
is called a zero of multiplicity m of f .
• ex. Form a polynomial whose zeros and degree are given below. Leave your
answer in factored form.
Zeros: -2, multiplicity 2; 4, multiplicity 1; degree: 3
• Def: The turning points of a function f are the points where f changes
direction.
• If r is a zero of even multiplicity then r is also a turning point, meaning that
the graph of f touches the x-axis at r. If r is a zero of odd multiplicity then
r is not a turning point, meaning that the graph of f corsses through the
x-axis at r.
• Fact: If f is a polynomial function of degree n, then f has at most n − 1
turning points.
• The end behavior of a polynomial function f (x) = an xn + . . . + a0 is the
behavior of the function of large values of |x|; i.e. the end behavior is what
the graph does as x approaches ∞ or −∞. The graph of f resembles an xn
for large values of |x|.
• To graph a polynomial function which is in factored form, you do the following:
1. Figure out the x− and y- intercepts of f .
2. Divide the x-axis into intervals with the x-intercepts marking the ends
of the intervals.
4
3. In each interval, determine whether the graph is above or below the xaxis. To do this, pick any point in the interval, plug it into the function,
and determine whether the value of the function at that point is positive
or negative. If it is positive then the graph is above the x-axis and if it
is negative then the graph is below the x-axis.
4. For the intervals containing ∞ or −∞, use the end behavior of the function to determine what the graph looks like in these intervals.
• ex. Graph:
a) f (x) = x − x3
5
b) f (x) = x2 (x − 3)(x + 4)
c) f (x) = x2 (x2 + 1)(x + 4)
6
Section 5.1: Example Answers
• ex. Graph:
(a) f (x) = (x − 2)5
(b) f (x) = x4 + 2
1
(c) f (x) = 3 − (x + 2)4
• ex. Graph:
a) f (x) = x − x3
2
b) f (x) = x2 (x − 3)(x + 4)
c) f (x) = x2 (x2 + 1)(x + 4)
3
Section 5.2: Properties of Rational Functions
• Def: A rational function is a function of the form R(x) = p(x)
, where p(x)
q(x)
and q(x) are polynomial functions and q(x) is not the zero polynomial. The
domain of a rational function is the set of all real numbers except those for
which q(x) = 0.
• ex. Find the domain of f (x) =
• The graph of f (x) =
1
x2
• ex. Graph: f (x) = 1 −
−x2 −7x−12
.
2x2 +7x−4
looks like:
1
(x−3)2
1
• Def: A horizontal asymptote of the graph of a function f (x) is a horizontal
line y = L such that as x approaches ∞ or −∞, f (x) approaches L. Usually
we write this as “f (x) → L as x → ±∞” or limx→±∞ f (x) = L.
• Def: A vertical asymptote of the graph of a function f (x) is a vertical line
x = c such that as x approaches c, the value of f (x) approaches ∞ or −∞.
• A horizontal asymptote describes the end behavior of a function f . The graph
of a function f can intersect a horizontal asymptote but it can never intersect
the vertical asymptote.
• ex. Find the vertical and horizontal asymptotes of f (x) =
1
x−2
+ 1.
• Def: A slant or oblique asymptote is an asymptote which is neither vertical
or horizontal. It is a line y = mx + b which f approaches as x approaches ∞
or −∞.
• If a rational function R(x) = p(x)
is in lowest terms (meaning that p(x) and
q(x)
q(x) have no common factors) then R will have a vertical asymptote for each
value of x for which q(x) = 0.
• Fact: Let R(x) =
p(x)
q(x)
=
an xn +...+a0
bm xm +...+b0
be a rational function.
1. If the degree of q(x) is greater than the degree of p(x) (so m > n) then
the line y = 0 is a horizontal asymptote of the graph of R.
2. If the degree of q(x) equals the degree of p(x) (so m = n) then the line
y = abnn is a horizontal asymptote of the graph of R.
3. If the degree of q(x) is one less than the degree of p(x) (so m = n−1) then
use long division to rewrite R(x) as R(x) = mx + b + r(x), where r(x) is
the remainder you get from long division. Then the line y = mx + b is
the oblique asymptote.
4. In all other cases, there are no horizontal or oblique asymptotes.
• ex. Find the vertical, horizontal, and oblique asymptotes, if any, of the
following rational functions:
2
a) G(x) =
−x2 +1
x+5
b) F (x) =
−2x2 +1
2x3 +4x2
3
Section 5.2: Example Answers
• ex. Graph: f (x) = 1 −
1
(x−3)2
• ex. Find the vertical and horizontal asymptotes of f (x) =
Vertical Asymptote: x = 2
Horizontal asymptote: y = 1
1
1
x−2
+ 1.
Section 5.3: The Graph of a Rational Function
• Recall that if R(x) = p(x)
is a rational function in lowest terms, then to say
q(x)
that x = r is a vertical asymptote is equivalent to saying that q(r) = 0 (so r
is a zero of q(x)), which is equivalent to saying that (x − r) is a factor of q(x).
If R(x) is not in lowest terms and (x − r) is a factor common in both the
numerator and denominator, then x = r is not a vertical asymptote. Instead,
the point x = r is a point not in the domain of the function and is represented
graphically as a hole in the function.
• If r is a zero of even multiplicity of q(x) then the graph will head in the
same direction on both sides of the vertical asymptote. If r is a zero of odd
multiplicity of q(x) then the graph will head in opposite directions on either
side of the vertical asymptote.
• To graph a rational function, R(x) =
p(x)
,
q(x)
use the following procedure:
1. Factor the numerator and denominator of R and find the domain of R
by finding the zeros of q.
2. Determine whether the graph has a hole or vertical asymptote at each
point not in the domain of R. For the zeros of q which correspond to
vertical asymptotes, determine whether they are zeros of even or odd
multiplicity.
3. Cancel any factors common in both the numerator and denominator so
that R is in lowest terms.
4. Locate the x- and y-intercepts of R. For any x-intercepts, determine
whether the point is a zero of even or odd multiplicity.
5. Test for symmetry. (This is not essential, but can be helpful.)
6. Locate any horizontal or oblique asymptotes. If there are none, determine
the end behavior by looking at the resulting polynomial obtained by long
dividing.
7. Use the x-intercepts and vertical asymptotes to divide the x-axis into
smaller intervals. Determine whether the graph is above or below the
x-axis in each of these intervals by picking any value of x within each
interval and determining if the corresponding y-value is positive or negative.
8. Graph the asymptotes, intercepts, and holes then use the rest of the
information to determine the rest of the graph.
1
• ex. Graph:
(a) R(x) =
x2 +3x−10
x2 +8x+15
(b) G(x) =
x3 +1
x2 +2x
2
(c) R(x) =
6
x2 −x−6
(d) Q(x) =
x4 −1
x2 −4
3
(e) R(x) =
x2 +x−12
x2 −4
(f) f (x) = 2x +
1
2x3
4
• ex. Find a possible rational function that might have the given graph:
5
Section 5.3: Example Answers
• ex. Graph:
a) R(x) =
x2 +3x−10
x2 +8x+15
b) G(x) =
x3 +1
x2 +2x
1
c) R(x) =
6
x2 −x−6
d) Q(x) =
x4 −1
x2 −4
2
e) R(x) =
x2 +x−12
x2 −4
f) f (x) = 2x +
1
2x3
3
Section 5.4: Polynomial and Rational Inequalities
• To solve a polynomial inequality, use the following procedure:
1. Get zero on one side of the inequality.
2. Find the zeros of the polynomial.
3. Use the zeros you found in the previous step to divide the real line into
smaller intervals. In each interval, determine whether the function is
positive or negative by picking any number in the interval and seeing if
the function is greater or less than zero at that point.
4. Figure out in which interval the inequality is true.
• ex. Solve the following:
a) x2 − 1 < 0
b) 2(2x2 − 3x) > −9
1
c) (x + 2)(x + 4)(x + 6)2 ≥ 0
d) x3 > 1
• To solve a rational inequality, use the following procedure:
1. Get zero on one side of the inequality and write the other side as a single
fraction if it isn’t already written that way.
2. Find the zeros of the numerator and denominator.
3. Use the zeros you found in the previous step to divide the real line into
smaller intervals. In each interval, determine whether the function is
positive or negative. To do this, pick any number in the interval and
determine whether each factor in each the numerator and denominator
is greater or less than zero at that point, then use that to figure out
whether the entire fraction is greater or less than zero.
4. Figure out in which interval the inequality is true.
2
• ex. Solve the following:
a) x −
b)
x+8
x+5
c)
3x+8
x+2
15
x
<2
≥1
≥1
3
d)
2x(x2 +4)(x−4)
(x+2)(x−2)
≥0
• ex. Find the domain of the function:
√
a) f (x) = x4 − 16.
b) f (x) =
q
x−1
.
x+4
4
Section 5.5: The Real Zeros of a Polynomial Function
• Remainder Theorem: Let f be a polynomial function and c be a real number.
If f (x) is divided by x − c, then the remainder is f (c).
• Factor Theorem: Let f be a polynomial function and c be a real number.
Then either both x − c is a factor of f (x) and f (c) = 0 or neither x − c is a
factor of f (x) nor f (c) = 0.
• If you combine the Remainder Theorem with the Factor Theorem, you get
the statement that either both x − c is a factor of f (x) and the remainder
when f (x) is divided by x − c is zero or neither x − c is a factor of f (x) nor
the remainder when f (x) is divided by x − c is zero.
• ex. Determine whether x − c is a factor of f (x). If it is not, then determine
the remainder of dividing f (x) by x − c.
a) f (x) = x4 − 8x2 − 9; x − 3
b) f (x) = 2x3 − x2 − 5x − 2; x − 1
• ex. Find k such that f (x) = x3 + kx2 − 2kx − 8 has the factor x + 1.
1
• Theorem: A polynomial function cannot have more real zeros than its degree.
• Decartes’ Rule of Signs: Let f be a polynomial function written in standard
form (meaning not factored and written in descending order of the exponents).
1. The number of positive real zeros of f either equals the number of sign
changes of consecutive nonzero coefficients of f (x) or else equals that
number minus an even integer.
2. The number of negative real zeros of f either equals the number of sign
changes of consecutive nonzero coefficients of f (−x) or else equals that
number minus an even integer.
• ex. Find the maximum number of zeros the polynomial function may have.
Then use Decartes’ Rule of Signs to determine how many positive and how
many negative zeros each function may have.
a) f (x) = −2x4 + 4x3 + 7
b) f (x) = −2x3 − 2x2 + 5x + 1
c) f (x) = 3x6 + 2x5 − 4x4 + 3x3 − 4x2 + 5x − 3
• Rational Zeros Theorem: Let f be a polynomial function of degree 1 or higher
of the form:
f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0
where an and a0 are not zero and all the coefficients are integers. Then if f
has any rational zeros then they will be of the form pq , where p is a factor of
a0 and q is a factor of an .
2
• ex. List the potential factors of the polynomial function.
a) f (x) = x5 − 4x3 + 3x2 − 5
b) f (x) = 3x4 − 2x3 − 2x2 + 1
c) f (x) = −4x4 − 3x2 + x + 6
• Theorem: Every polynomial function with real coefficients can be uniquely
factored into a product of linear factors (that is, factors of the form x − c),
and/or irreducible quadratic factors (that is, factors of the form ax2 + bx + c
which cannot be factored).
• As a consequence of the previous theorem, a polynomial function with real
coefficients of odd degree has at least one real zero.
• To find the real zeros of a polynomial function, use the following procedure:
1. Use the degree of the polynomial to determine the maximum number of
real zeros.
2. Use Decartes’ Rule of Signs to determine the possible number of positive
and negative zeros.
3. If the polynomial has integer coefficients, use the Rational Zeros Theorem
to identify the potential rational zeros. Then test to see if any are zeros.
4. If you locate a zero, call it c, divide f by x − c to factor f . Then repeat
the previous step on the remaining factor of f to determine any other
zeros.
3
5. Repeat the previous step until you have found all the zeros.
• ex. Find all real zeros of the polynomial function. Then use the zeros to
factor f over the real numbers.
a) f (x) = x3 + 6x2 − x − 30
b) f (x) = 3x4 + 9x3 − 3x2 − 27x − 18
c) f (x) = x4 − x3 − 7x2 + 13x − 6
• ex. Solve the equation in the real number system: x3 +
4
11 2
x
2
+ 12x + 9 = 0
Section 5.6: Complex Zeros; Fundamental Theorem of
Algebra
• Def: A complex polynomial function f of degree n is a function of the form
f (x) = an xn + an−1 xn−1 + . . . + a1 x + a0
where an , . . . , a0 are complex numbers and all the exponents are nonnegative
integers. A complex number r for which f (r) = 0 is called a complex zero of
f or a complex root of f .
• Fundamental Theorem of Algebra: Every complex polynomial function f (x)
of degree n ≥ 1 has at least one complex zero.
• Theorem: Every complex polynomial function f (x) of degree n ≥ 1 can be
factored into n linear factors (some of the factors may be repeated) of the
form
f (x) = an (x − r1 )(x − r2 ) . . . (x − rn )
where an , ra , . . . , rn are complex numbers. In other words, every complex
polynomial function of degree n ≥ 1 has exactly n zeros (although some
might be repeated). In fact, r1 , . . . , rn are the n zeros of f .
• Conjugate Pairs Theorem: Let f (x) be a polynomial whose coefficients are
real numbers. If r = a+bi is a zero of f , then the complex conjugate r = a−bi
is also a zero of f .
• As a consequence of this theorem, we can conclude that a polynomial f of
odd degree with real coefficients has at least one real zero.
• ex. Information is given about a polynomial f (x) whose coefficients are real
numbers. Find the remaining zeros.
a) Degree 4; zeros: 0, −1, 4 − 3i
b) Degree 6; zeros: 4i, 5 − 2i, −6 + 2i
• Form a polynomial f (x) with real coefficients having the given degree and
zeros.
a) Degree 4; zeros: −2i, 2 + 3i
1
b) Degree 5; zeros: 2, multiplicity 3; 1 − i
• Use the given zero to find the remaining zeros of the function.
a) g(x) = x3 + 3x2 + 25x + 75; zero: −5i
b) h(x) = 4x4 − 12 + 13x2 − x3 − 4x; zero: 2i
2
• Find the complex zeros of the polynomial function. Write f in factored form.
a) f (x) = x3 + 13x2 + 57x + 85
b) f (x) = x4 + 13x2 + 36
c) f (x) = x4 + 3x3 − 19x2 + 27x − 252
3