4. Polynomial and Rational Functions Flashcards

1
Q

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 _______.

A

standard

parabola

(h, k)

> 0

< 0

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2
Q

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 ______.

A

– b / 2a

f(– b / 2a)

– b / 2a

f(– b / 2a)

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3
Q

True or False

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

A

True

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4
Q

True or False

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

A

False

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5
Q

True or False

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

A

True

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6
Q

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) = ______.

A

x – 8

x² – 8x

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7
Q

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) = ______.

A

40 – x

–x² + 40x

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8
Q

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

A

5

–2

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9
Q

True or false:

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

A

False

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10
Q

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.

A

end

leading

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11
Q

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

A

falls

rises

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12
Q

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

A

rises

falls

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13
Q

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

A

rises

rises

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14
Q

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

A

falls

falls

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15
Q

True or false:

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

A

true

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16
Q

True or false:

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

A

true

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17
Q

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

A

x-intercept

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18
Q

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.

A

turns around

crosses

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19
Q

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.

A

0

Intermediate Value

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20
Q

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

21
Q

Consider the following long division problem:

x + 4⟌(6x – 4 + 2x^3)

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

A

2x^3 + 0x² + 6x – 4

22
Q

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.

A

6x^3

3x

2x²

7x²

23
Q

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 ______.

A

2x²

5x – 2

10x^3 – 4x²

10x^3 + 6x²

24
Q
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.

A

6x² – 10x

6x² + 8x

18x

–4

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 ______.
9 3x – 5 9 3x – 5 + 9/(2x + 1)
26
After performing polynomial long division, the answer may be checked by multiplying the _____ by the _____, and then adding the _____. You should obtain the ______.
divisor quotient remainder dividend
27
To divide x^3 + 5x² – 7x + 1 by x – 4 using the synthetic division, the first step is to write _ | _ _ _ _
4 1 5 –7 1
28
To divide 4x^3 – 8x – 2 by x + 5 using synthetic division, the first step is to write _ | _ _ _ _
–5 4 0 –8 –2
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)
True
30
The Remainder Theorem states that if the polynomial f(x) is divided by x – c, then the remainder is ______.
f(c)
31
The Factor Theorem states that if f is a polynomial function and f(c) = 0, then _____ is a factor of f(x).
x – c
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 _____.
the constant the leading coefficient
33
True or false 3/2 is a possible rational zero of f(x) = 2x^3 + 11x² – 7x – 6.
true
34
True or false 1/2 is a possible rational zero of f(x) = 3x^4 – 3x^3 + x² – x + 1
false
35
If a polynomial equation is of degree n, then counting multiple roots separately, the equation has _____ roots.
n
36
If a + bi is a root of a polynomial equation with real coefficients, b ≠ 0, then _____ is also a root of the equation.
a – bi
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 ______.
–6 (x + 6) (2x² – x – 1) = 0
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 _____.
n 1
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.
false
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.
true
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.
true
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: _____, _____, _____.
x² + 8x + 15 = 0 boundary (–∞, –5) (–5, –3) (–3, ∞)
43
y varies directly as x can be modeled by the equation _____, where k is called the _______.
y = kx constant of variation
44
y varies directly as the nth power of x can be modeled by the equation _______.
y = kx^n
45
y varies inversely as x can be modeled by the equation _____.
y = k / x
46
y varies directly as x and inversely as z can be modeled by the equation _______.
y = kx / z
47
y varies jointly as x and z can be modeled by the equation ______.
y = kxz
48
In the equation S = 8A/P, S varies _____ as A and ______ as P.
directly inversely
49
In the equation C = (0.02P1P2) / D², C varies _____ as P1 and P2 and _____ as the square of d.
jointly inversely