Chapter Two: Polynomial and Rational Functions

Question 1

divisor

Answer 1

the number on the outside of the division sign

Question 2

dividend

Answer 2

the number inside of the division sign

Question 3

quotient

Answer 3

the answer on top of the division sign

Question 4

remainder

Answer 4

the number left over at the bottom of the long division

Question 5

the dividend is equal to

Answer 5

the divisor times the quotient plus the remainder (divisor)(quotient) + remainder

Question 6

if a question asks, “the quotient of x and y . . .” which is the divisor and which is the dividend?

Answer 6

x is the dividend (inside the division sign) and y is the divisor (outside the division sign)

Question 7

the remainder theorem

Answer 7

if a polynomial p(x) is divided by (x-c), then its remainder is equal to p(c)

Question 8

the factor theorem

Answer 8

the polynomial (x-k) is a factor of a polynomial f(x) if and only if f(k) = 0

Question 9

the rational zero theorem

Answer 9

f(x) = a_nXⁿ + a_n-1X^n-1 + . . . + a₁X + a₀

if p/q is a zero of f(x) then p is a factor of a₀ and q is a factor of a_n

in english: if you have a fraction as a zero for a polynomial, the numerator will be a factor of the constant, and the denominator will be a factor of the coefficient of the highest degree

Question 10

synthetic division only works when the divisor is in the form

Answer 10

( x +/- c)

Question 11

if given one zero of a polynomial, the other zeroes can be found using:

Answer 11

synthetic division, then factoring

Question 12

what formula can be used to find the zeroes of polynomials?

Answer 12

f(x) = a(x - c)(x - c)(x - c)

Question 13

multiplicity

Answer 13

how many times a given root occurs

ex: (x - 3)² –> the root x = 3 has a multiplicity of 2

Question 14

to find the multiplicity of the roots of a cubic:

Answer 14

1) factor out a common factor to create a binomial that can be factored, or use the sum or difference of two cubes
2) if you are only given the quadratic and no roots, use the rational zero theorem to find the zeroes

Question 15

difference of two cubes

Answer 15

a³ - b³ = (a - b)(a² + ab + b²)

Question 16

sum of two cubes

Answer 16

a³ + b³ = (a + b)(a² - ab + b²)

Question 17

Descartes Rule of Sign

Answer 17

the number of real roots (positive or negative) is either equal to the number of sign changes or the number of sign changes minus an even number

• to find positive roots, evaluate f(x)
• to find negative roots, evaluate f(-x)

Question 18

the maximum number of regions on the graph of a polynomial is determined by

Answer 18

the highest power of the polynomial

Question 19

the maximum number of turning points on the graph of a polynomial is

Answer 19

one less than the polynomial’s degree

Question 20

if the degree is even the graph of a polynomial will be similar to a

Answer 20

parabola

Question 21

if the degree is odd the graph of the polynomial will be similar to

Answer 21

the graph of x³

Question 22

for even degrees, as the degree increases, the parabola becomes

Answer 22

more steep and u-shaped

Question 23

all even degree graphs share the points

Answer 23

(1, 1) and (-1, 1)

Question 24

for odd degrees, as the degree increases, the function becomes

Answer 24

tighter

Question 25

all odd degree graphs share the points

Answer 25

(1, 1) and (-1, -1)

Question 26

if x = r is a zero of polynomial P(x) with multiplicity k then:

Answer 26

1) if k is odd the x-intercept will cross the x-axis
2) if k is even the x-intercept will touch the x-axis
3) if k > 1, the graph will flatten out at x = r

Question 27

rational function

Answer 27

the quotient of two polynomials

Question 28

asymptote

Answer 28

a line the function approaches but never touches

• the function can cross a horizontal asymptote because horizontal asymptotes are more “guidelines”
• the function cannot cross a vertical asymptote because a vertical asymptote is a solid “rule,” and the function is undefined at that value

Question 29

to find a vertical asymptote (x = c)

Answer 29

1) cancel common factors
2) set the denominator equal to zero and solve
3) the result is the vertical asymptote

Question 30

if the denominator of a function cancels out completely, then

Answer 30

there is no vertical asymptote, but instead there is a hole in the graph

Question 31

to find the horizontal asymptote (y = c)

Answer 31

1) if the degree of the denominator is higher than the numerator –> y = 0 (the x-axis)
2) if the degree of the numerator is higher than the denominator –> no HA
3) if the degrees are the same –> y = a₁/a₂ (the quotient of the leading coefficients)

Question 32

oblique/slant asymptotes occur when

Answer 32

the degree of the numerator is one larger than the degree of the denominator

Question 33

to find the slant/oblique asymptote

Answer 33

1) reduce
2) long divide the polynomials
3) the quotient is the slant asymptote

Question 34

rational inequality

Answer 34

an inequality that contains one or more rational expressions

Question 35

how do you solve a rational inequality?

Answer 35

1) set the rational expression in relation to 0 (the inequality MUST have 0 on the right!!)
2) find the critical values
a. set the numerator equal to 0 and solve
b. set the denominator equal to 0 and solve
c. factor if possible
3) mark the critical values on a number line, dividing it into regions
4) test a value from each region in the inequality. If the result is true, that entire region is true.
5) write the solution using inequalities, interval notation, or set notation

Question 36

º and •

Answer 36

º = not a solution

• = solution

Question 37

multiplying or dividing an inequality be a negative does what?

Answer 37

flips the sign of the inequality

Question 38

domain

Answer 38

the collection of x’s that yield a real number