4. Polynomial and Rational Functions

Question 1

The quadratic function f(x) = a(x – h)² + k, a ≠ 0, is in ______ form.

The graph of f is called a/an ______ whose vertex is the point _______.

The graph opens upward if a ______ and opens downwards if a _______.

Answer 1

standard

parabola

(h, k)

> 0

< 0

Question 2

Consider the quadratic function f(x) = ax² + bx + c, a ≠ 0.

If a > 0, then f has a minimum that occurs at x = ______. This maximum value is _______.

if a < 0, then f has a maximum that occurs at x = _____. This maximum value is ______.

Answer 2

– b / 2a

f(– b / 2a)

– b / 2a

f(– b / 2a)

Question 3

True or False

The graph of f(x) = (x – 2)² + 1 opens upward.

Answer 3

True

Question 4

True or False

The graph of f(x) = (x + 5)² + 3 has its vertex (5, 3).

Answer 4

False

Question 5

True or False

The y-coordinate of the vertex of f(x) = 4x² – 16x + 300 is f(2).

Answer 5

True

Question 6

The difference between two numbers is 8. If one number is represented by x, the other number can be expressed as _______.

The product of the numbers, P(x), expressed in the form P(x) = ax² + bx + c, is P(x) = ______.

Answer 6

x – 8

x² – 8x

Question 7

The perimeter of a rectangle is 80 feet. If the length of the rectangle is represented by x, its width can be expressed as ______. The area of the rectangle, A(x), expressed in the form A(x) = ax² + bx + c, is A(x) = ______.

Answer 7

40 – x

–x² + 40x

Question 8

The degree of the polynomial function f(x) = –2x^3(x –1)(x + 5) is _____. The leading coefficient is _____.

Answer 8

5

–2

Question 9

True or false:

Some polynomial functions of degree 2 or higher have breaks in their graphs.

Answer 9

False

Question 10

The behavior of the graph of a polynomial function to the far left or the far right is called its _____ behavior, which depends upon the ______ term.

Answer 10

end

leading

Question 11

The graph of f(x) = x^3 _____ to the left and ______ to the right.

Answer 11

falls

rises

Question 12

The graph of f(x) = –x^3 _____ to the left and ______ to the right.

Answer 12

rises

falls

Question 13

The graph of f(x) = x² _____ to the left and ______ to the right.

Answer 13

rises

rises

Question 14

The graph of f(x) = –x² ______ to the left and ______ to the right.

Answer 14

falls

falls

Question 15

True or false:

Odd-degree polynomial functions have graphs with opposite behavior at each end.

Answer 15

true

Question 16

True or false:

Even-degree polynomial functions have graphs with the same behavior at each end.

Answer 16

true

Question 17

Every real zero of a polynomial function appears as a/an ______ of the graph.

Answer 17

x-intercept

Question 18

If r is a zero of even multiplicity, then the graph touches the x-axis and _____ at r. If r is a zero of odd multiplicity, then the graph ______ the x-axis at r.

Answer 18

turns around

crosses

Question 19

If f is a polynomial function and f(a) and f(b) have opposite signs, then there must be at least one value of c between a and b for which f(c) = ______. This result is called the ______ Theorem.

Answer 19

0

Intermediate Value

Question 20

If f is a polynomial function of degree n, then the graph of f has at most ______ turning points.

Answer 20

n – 1

Question 21

Consider the following long division problem:

x + 4⟌(6x – 4 + 2x^3)

We begin the division process by rewriting the dividend as ______.

Answer 21

2x^3 + 0x² + 6x – 4

Question 22

Consider the following long division problem:

3x – 1⟌(6x^3 + 7x² + 12x –5)

We begin the division process by dividing ______ by ______.

We obtain _____.

We write this result above _____ in the dividend.

Answer 22

6x^3

3x

2x²

7x²

Question 23

In the following long division problem, the first step has been completed:

                       2x² 5x – 2⟌(10x^3 + 6x² – 9x + 10)

The next step is to multiply _____ and _____.

We obtain _____.

We write this result below ______.

Answer 23

2x²

5x – 2

10x^3 – 4x²

10x^3 + 6x²

Question 24

In the following long division problem, the first two steps have been completed:
                       2x
3x – 5⟌(6x² + 8x – 4)
             6x² – 10x
             --------------

The next step is to subtract _____ from ______.

We obtain _____.

Then we bring down _____ and form the new dividend.

Answer 24

6x² – 10x

6x² + 8x

18x

–4

Question 25

In the following long division problem, most of the steps have been completed:
                      3x – 5
2x + 1⟌(6x² – 7x + 4)
            6x² + 3x
            ------------
                 – 10x + 4
                 – 10x – 5
                ---------------
                               ?

Completing the step designated by the question mark, we obtain ______.

Thus, the quotient is _____ and the remainder is ______.

The answer to this long division problem is ______.

Answer 25

9

3x – 5

9

3x – 5 + 9/(2x + 1)

Question 26

After performing polynomial long division, the answer may be checked by multiplying the _____ by the _____, and then adding the _____.

You should obtain the ______.

Answer 26

divisor

quotient

remainder

dividend

Question 27

To divide x^3 + 5x² – 7x + 1 by x – 4 using the synthetic division, the first step is to write

_ | _ _ _ _

Answer 27

4

1

5

–7

1

Question 28

To divide 4x^3 – 8x – 2 by x + 5 using synthetic division, the first step is to write

_ | _ _ _ _

Answer 28

–5

4

0

–8

–2

Question 29

True or false:

1 | 3 –4 2 –1
–3 7 –9
———————–
3 –7 9 –10

means

(3x^3 – 4x² + 2x – 1) / (x + 1) = 3x² – 7x + 9 – 10 / (x + 1)

Answer 29

True

Question 30

The Remainder Theorem states that if the polynomial f(x) is divided by x – c, then the remainder is ______.

Answer 30

f(c)

Question 31

The Factor Theorem states that if f is a polynomial function and f(c) = 0, then _____ is a factor of f(x).

Answer 31

x – c

Question 32

The Rational Zero Theorem states that if p/q is a rational zero of f (where p/q is reduced to lowest terms), then p is a factor of _____ and q is a factor of _____.

Answer 32

the constant

the leading coefficient

Question 33

True or false

3/2 is a possible rational zero of f(x) = 2x^3 + 11x² – 7x – 6.

Answer 33

true

Question 34

True or false

1/2 is a possible rational zero of f(x) = 3x^4 – 3x^3 + x² – x + 1

Answer 34

false

Question 35

If a polynomial equation is of degree n, then counting multiple roots separately, the equation has _____ roots.

Answer 35

n

Question 36

If a + bi is a root of a polynomial equation with real coefficients, b ≠ 0, then _____ is also a root of the equation.

Answer 36

a – bi

Question 37

Consider solving 2x^3 + 11x² – 7x – 6 = 0. The synthetic division shown below indicates that _____ is a root.

–6| 2 11 –7 –6
–12 6 6
———————–
2 –1 –1 0

Based on the synthetic division, 2x^3 + 11x² – 7x – 6 = 0 can be written in factored form as ______.

Answer 37

–6

(x + 6) (2x² – x – 1) = 0

Question 38

The Linear Factorization Theorem states that an nth-degree polynomial can be expressed as the product of a nonzero constant and _____ linear factors, where each linear factor has a leading coefficient of _____.

Answer 38

n

1

Question 39

Use Descartes’s Rule of Signs to determine whether the statement is true or false.

A polynomial function with four signs changes must have four positive real zeros.

Answer 39

false

Question 40

Use Descartes’s Rule of Signs to determine whether the statement is true or false.

A polynomial function with one sign change must have one positive real zero.

Answer 40

true

Question 41

Use Descartes’s Rule of Signs to determine whether the statement is true or false.

A polynomial function with seven sign changes can have one, three, five, or seven positive real zeros.

Answer 41

true

Question 42

We solve the polynomial inequality x² + 8x + 15 > 0 by first solving the equation ______.

The real solutions of this equation, –5 and –3, are called ______ points.

The points at –5 and –3 divide the number line into three intervals: _____, _____, _____.

Answer 42

x² + 8x + 15 = 0

boundary

(–∞, –5)

(–5, –3)

(–3, ∞)

Question 43

y varies directly as x can be modeled by the equation _____, where k is called the _______.

Answer 43

y = kx

constant of variation

Question 44

y varies directly as the nth power of x can be modeled by the equation _______.

Answer 44

y = kx^n

Question 45

y varies inversely as x can be modeled by the equation _____.

Answer 45

y = k / x

Question 46

y varies directly as x and inversely as z can be modeled by the equation _______.

Answer 46

y = kx / z

Question 47

y varies jointly as x and z can be modeled by the equation ______.

Answer 47

y = kxz

Question 48

In the equation S = 8A/P, S varies _____ as A and ______ as P.

Answer 48

directly

inversely

Question 49

In the equation C = (0.02P1P2) / D², C varies _____ as P1 and P2 and _____ as the square of d.

Answer 49

jointly

inversely