Polynomial & Rational Functions

Synthetic Division

Left and Right Behavior of a Polynomial

1.a_n> 0 and n even
Graph of P(x) increases without
bound as x decreases to the left
and as x increases to the right.

P(x) →∞as x→–
P(x) →∞as x→∞

2.a_n> 0 and n odd
Graph of P(x) decreases without
bound as x decreases to the left
and increases without bound as x
increases to the right.

P(x) →–∞as x→–∞
P(x) →∞as x→∞

3.a_n< 0 and n even
Graph of P(x) decreases without
bound as x decreases to the left
and as x increases to the right.

P(x) →–∞as x→–∞
P(x) →–∞as x→∞

4.a_n< 0 and n odd
Graph of P(x) increases without
bound as x decreases to the left and
decreases without bound as x
increases to the right.

P(x) →∞as x→–∞
P(x) →–∞as x→∞

Fundamental Theorem of Algebra

Every polynomial P (x) of degree n> 0 has at least one zero .

n Zeros Theorem

Every polynomial P(x) of degree n> 0 can be ex pressed as the product
of n linear factors . Hence, P(x) has exactly n zeros—not necessarily distinct.

Imaginary Zeros Theorem

Imaginary zeros of polynomials with real coefficients , if they exist, occur
in conjugate pairs.

Real Zeros and Odd-Degree Polynomials

A polynomial of odd degree with real coefficients always has at least
one real zero.

Strategy for Finding Rational Zeros

Assume that P(x) is a polynomial with integer coefficients and is of degree greater
than 2.

Step 1.List the possible rational zeros of P(x) using the rational zero theorem
(Theorem 6).

Step 2.Construct a synthetic division table . If a rational zero r is found, stop, write

P(x) = (x–r)Q(x)

and immediately proceed to find the rational zeros for Q(x), the reduced
polynomial relative to P(x). If the degree of Q(x) is greater than 2,
return to step 1 using Q (x) in place of P(x). If Q(x) is quadratic,
find all its zeros using standard methods for solving quadratic equations.

Location Theorem

If fis continuous on an interval I, aand bare two numbers in I , and f(a)
and f(b) are of opposite sign, then there is at least one xintercept between aand b.

Upper and Lower Bounds of Real Zeros

Given an nth-degree polynomial P(x) with real coefficients, n> 0, a_n> 0,
and P(x) divided by x–r using synthetic division:

1.Upper Bound.If r> 0 and all numbers in the quotient row of the synthetic division,
including the remainder, are nonnegative, then r is an upper bound of the real
zeros of P(x).

2.Lower Bound.If r< 0 and all numbers in the quotient row of the synthetic division,
including the remainder, alternate in sign, then r is a lower bound of the real
zeros of P(x).

[Note: In the lower-bound test, if 0 appears in one or more places in the quotient row,
including the remainder, the sign in front of it can be considered either positive or
negative , but not both. For example, the numbers 1, 0, 1 can be considered to alternate
in sign, while 1, 0, –1 cannot.]
 

The Bisection Method

Approximate to one decimal place the zero o

P(x) = x⁴–2x³–10x²+ 40x–9

on the interval (3, 4).

Nested intervals produced by the Bisection Method

Synthetic Division on a Graphing Utility

Program SNYDIV

OUTPUT

Graphing a Rational Function:

Step1.Intercepts. Find the real solutions of the equation n (x) = 0 and use
these solutions to plot any x intercepts of the graph of f.
Evaluate f(0), if it exists, and plot the y intercept.

Step 2.Vertical Asymptotes. Find the real solutions of the equation d(x) = 0
and use these solutions to de termine the domain of f, the points of
discontinuity, and the vertical asymptotes. Sketch any vertical
asymptotes as dashed lines.

Step3.Sign Chart. Construct a sign chart for f and use it to determine the
behavior of the graph near each vertical asymptote.

Step 4.Horizontal Asymptotes. Determine whether there is a horizontal
asymptote and if so, sketch it as a dashed line.

Step5.Symmetry.Determine symmetry with respect to the vertical axis and
the origin.

Step 6.Complete the Sketch. Complete the sketch of the graph by plotting
additional points and joining these points with a smooth continuous
curve over each interval in the domain of f. Do not cross any points
of discontinuity.

Partial Fraction Decomposition

Any proper fraction P(x)/D(x) reduced to lowest terms can be decomposed in the sum of partial fractions as follows

1.If D(x) has a nonrepeating linear factor of the form ax+ b, then the partial fraction decomposition of P(x)/D(x) contains a term of the form
A a constant

2.If D(x) has a k-repeating linear factor of the form (ax+ b)^k, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form

3.If D(x) has a nonrepeating quadratic factor of the form ax²+ bx+ c, which is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains a term of the form

4.If D(x) has a k-repeating quadratic factor of the form (ax²+ bx+ c)^k, where ax²+ bx+ cis prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains k terms of