Section 5.1: Polynomial Functions

• Def: A polynomial function is a function of the form:

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0

where an, an−1, . . . , a1, a0 are real numbers and the exponents are all non-negative integers. The domain of a polynomial function is the set of all realnumbers.

• Def: The degree of a polynomial function is the largest exponent of x thatappears 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 holesin 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 evenor odd.

1. n is even

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2. n odd

• ex. Graph:

(a) f(x) = (x− 2)5

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(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 aresometimes 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 numbersuch that f(r) = 0, then r is a real zero of f . Conversely, if r is a zero of

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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 youranswer 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 calleda repeated zero of f or a multiple zero of f . More precisely, if (x− r)m is afactor of a polynomial function f and (x− r)m+1 is not a factor of f , then ris called a zero of multiplicity m of f .

• ex. Form a polynomial whose zeros and degree are given below. Leave youranswer 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 changesdirection.

• If r is a zero of even multiplicity then r is also a turning point, meaning thatthe graph of f touches the x-axis at r. If r is a zero of odd multiplicity thenr is not a turning point, meaning that the graph of f corsses through thex-axis at r.

• Fact: If f is a polynomial function of degree n, then f has at most n − 1turning points.

• The end behavior of a polynomial function f(x) = anxn + . . . + a0 is the

behavior of the function of large values of |x|; i.e. the end behavior is whatthe graph does as x approaches ∞ or −∞. The graph of f resembles anx

n

for large values of |x|.

• To graph a polynomial function which is in factored form, you do the follow-ing:

1. Figure out the x− and y- intercepts of f .

2. Divide the x-axis into intervals with the x-intercepts marking the endsof the intervals.

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3. In each interval, determine whether the graph is above or below the x-axis. 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 positiveor negative. If it is positive then the graph is above the x-axis and if itis negative then the graph is below the x-axis.

4. For the intervals containing∞ or −∞, use the end behavior of the func-tion to determine what the graph looks like in these intervals.

• ex. Graph:

a) f(x) = x− x3

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b) f(x) = x2(x− 3)(x + 4)

c) f(x) = x2(x2 + 1)(x + 4)

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Section 5.1: Example Answers

• ex. Graph:

(a) f(x) = (x− 2)5

(b) f(x) = x4 + 2

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(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)

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Section 5.2: Properties of Rational Functions

• Def: A rational function is a function of the form R(x) = p(x)q(x)

, where p(x)

and q(x) are polynomial functions and q(x) is not the zero polynomial. Thedomain of a rational function is the set of all real numbers except those forwhich q(x) = 0.

• ex. Find the domain of f(x) = −x2−7x−122x2+7x−4

.

• The graph of f(x) = 1x2 looks like:

• ex. Graph: f(x) = 1− 1(x−3)2

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• Def: A horizontal asymptote of the graph of a function f(x) is a horizontalline y = L such that as x approaches ∞ or −∞, f(x) approaches L. Usuallywe 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 linex = 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 graphof a function f can intersect a horizontal asymptote but it can never intersectthe vertical asymptote.

• ex. Find the vertical and horizontal asymptotes of f(x) = 1x−2

+ 1.

• Def: A slant or oblique asymptote is an asymptote which is neither verticalor horizontal. It is a line y = mx + b which f approaches as x approaches ∞or −∞.

• If a rational function R(x) = p(x)q(x)

is in lowest terms (meaning that p(x) and

q(x) have no common factors) then R will have a vertical asymptote for eachvalue of x for which q(x) = 0.

• Fact: Let R(x) = p(x)q(x)

= anxn+...+a0

bmxm+...+b0be a rational function.

1. If the degree of q(x) is greater than the degree of p(x) (so m > n) thenthe 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 liney = an

bnis 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) thenuse long division to rewrite R(x) as R(x) = mx + b + r(x), where r(x) isthe remainder you get from long division. Then the line y = mx + b isthe 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 thefollowing rational functions:

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a) G(x) = −x2+1x+5

b) F (x) = −2x2+12x3+4x2

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Section 5.2: Example Answers

• ex. Graph: f(x) = 1− 1(x−3)2

• ex. Find the vertical and horizontal asymptotes of f(x) = 1x−2

+ 1.

Vertical Asymptote: x = 2Horizontal asymptote: y = 1

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Section 5.3: The Graph of a Rational Function

• Recall that if R(x) = p(x)q(x)

is a rational function in lowest terms, then to say

that x = r is a vertical asymptote is equivalent to saying that q(r) = 0 (so ris 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 thenumerator 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 representedgraphically as a hole in the function.

• If r is a zero of even multiplicity of q(x) then the graph will head in thesame direction on both sides of the vertical asymptote. If r is a zero of oddmultiplicity of q(x) then the graph will head in opposite directions on eitherside 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 Rby finding the zeros of q.

2. Determine whether the graph has a hole or vertical asymptote at eachpoint not in the domain of R. For the zeros of q which correspond tovertical asymptotes, determine whether they are zeros of even or oddmultiplicity.

3. Cancel any factors common in both the numerator and denominator sothat R is in lowest terms.

4. Locate the x- and y-intercepts of R. For any x-intercepts, determinewhether 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, determinethe end behavior by looking at the resulting polynomial obtained by longdividing.

7. Use the x-intercepts and vertical asymptotes to divide the x-axis intosmaller intervals. Determine whether the graph is above or below thex-axis in each of these intervals by picking any value of x within eachinterval and determining if the corresponding y-value is positive or neg-ative.

8. Graph the asymptotes, intercepts, and holes then use the rest of theinformation to determine the rest of the graph.

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• ex. Graph:

(a) R(x) = x2+3x−10x2+8x+15

(b) G(x) = x3+1x2+2x

2

(c) R(x) = 6x2−x−6

(d) Q(x) = x4−1x2−4

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(e) R(x) = x2+x−12x2−4

(f) f(x) = 2x + 12x3

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• ex. Find a possible rational function that might have the given graph:

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Section 5.3: Example Answers

• ex. Graph:

a) R(x) = x2+3x−10x2+8x+15

b) G(x) = x3+1x2+2x

1

c) R(x) = 6x2−x−6

d) Q(x) = x4−1x2−4

2

e) R(x) = x2+x−12x2−4

f) f(x) = 2x+ 12x3

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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 intosmaller intervals. In each interval, determine whether the function ispositive or negative by picking any number in the interval and seeing ifthe 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

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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 singlefraction 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 intosmaller intervals. In each interval, determine whether the function ispositive or negative. To do this, pick any number in the interval anddetermine whether each factor in each the numerator and denominatoris greater or less than zero at that point, then use that to figure outwhether the entire fraction is greater or less than zero.

4. Figure out in which interval the inequality is true.

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• ex. Solve the following:

a) x− 15x

< 2

b) x+8x+5≥ 1

c) 3x+8x+2≥ 1

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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) =√

x−1x+4

.

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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 afactor of f(x) nor f(c) = 0.

• If you combine the Remainder Theorem with the Factor Theorem, you getthe statement that either both x − c is a factor of f(x) and the remainderwhen f(x) is divided by x− c is zero or neither x− c is a factor of f(x) northe 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 determinethe 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.

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• Theorem: A polynomial function cannot have more real zeros than its degree.

• Decartes’ Rule of Signs: Let f be a polynomial function written in standardform (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 signchanges of consecutive nonzero coefficients of f(x) or else equals thatnumber minus an even integer.

2. The number of negative real zeros of f either equals the number of signchanges of consecutive nonzero coefficients of f(−x) or else equals thatnumber 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 howmany 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 higherof the form:

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0

where an and a0 are not zero and all the coefficients are integers. Then if fhas any rational zeros then they will be of the form p

q, where p is a factor of

a0 and q is a factor of an.

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• 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 uniquelyfactored 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 + cwhich cannot be factored).

• As a consequence of the previous theorem, a polynomial function with realcoefficients 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 ofreal zeros.

2. Use Decartes’ Rule of Signs to determine the possible number of positiveand negative zeros.

3. If the polynomial has integer coefficients, use the Rational Zeros Theoremto 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 repeatthe previous step on the remaining factor of f to determine any otherzeros.

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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 tofactor 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 + 112x2 + 12x + 9 = 0

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Section 5.6: Complex Zeros; Fundamental Theorem ofAlgebra

• Def: A complex polynomial function f of degree n is a function of the form

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0

where an, . . . , a0 are complex numbers and all the exponents are nonnegativeintegers. A complex number r for which f(r) = 0 is called a complex zero off 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 befactored into n linear factors (some of the factors may be repeated) of theform

f(x) = an(x− r1)(x− r2) . . . (x− rn)

where an, ra, . . . , rn are complex numbers. In other words, every complexpolynomial function of degree n ≥ 1 has exactly n zeros (although somemight be repeated). In fact, r1, . . . , rn are the n zeros of f .

• Conjugate Pairs Theorem: Let f(x) be a polynomial whose coefficients arereal numbers. If r = a+bi is a zero of f , then the complex conjugate r = a−biis also a zero of f .

• As a consequence of this theorem, we can conclude that a polynomial f ofodd degree with real coefficients has at least one real zero.

• ex. Information is given about a polynomial f(x) whose coefficients are realnumbers. 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 andzeros.

a) Degree 4; zeros: −2i, 2 + 3i

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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

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• 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

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