4. Polynomial and Rational Functions Flashcards
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 _______.
standard
parabola
(h, k)
> 0
< 0
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 ______.
– b / 2a
f(– b / 2a)
– b / 2a
f(– b / 2a)
True or False
The graph of f(x) = (x – 2)² + 1 opens upward.
True
True or False
The graph of f(x) = (x + 5)² + 3 has its vertex (5, 3).
False
True or False
The y-coordinate of the vertex of f(x) = 4x² – 16x + 300 is f(2).
True
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) = ______.
x – 8
x² – 8x
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) = ______.
40 – x
–x² + 40x
The degree of the polynomial function f(x) = –2x^3(x –1)(x + 5) is _____. The leading coefficient is _____.
5
–2
True or false:
Some polynomial functions of degree 2 or higher have breaks in their graphs.
False
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.
end
leading
The graph of f(x) = x^3 _____ to the left and ______ to the right.
falls
rises
The graph of f(x) = –x^3 _____ to the left and ______ to the right.
rises
falls
The graph of f(x) = x² _____ to the left and ______ to the right.
rises
rises
The graph of f(x) = –x² ______ to the left and ______ to the right.
falls
falls
True or false:
Odd-degree polynomial functions have graphs with opposite behavior at each end.
true
True or false:
Even-degree polynomial functions have graphs with the same behavior at each end.
true
Every real zero of a polynomial function appears as a/an ______ of the graph.
x-intercept
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.
turns around
crosses
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.
0
Intermediate Value
If f is a polynomial function of degree n, then the graph of f has at most ______ turning points.
n – 1
Consider the following long division problem:
x + 4⟌(6x – 4 + 2x^3)
We begin the division process by rewriting the dividend as ______.
2x^3 + 0x² + 6x – 4
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.
6x^3
3x
2x²
7x²
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 ______.
2x²
5x – 2
10x^3 – 4x²
10x^3 + 6x²
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.
6x² – 10x
6x² + 8x
18x
–4
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)
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
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
To divide 4x^3 – 8x – 2 by x + 5 using synthetic division, the first step is to write
_ | _ _ _ _
–5
4
0
–8
–2
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
The Remainder Theorem states that if the polynomial f(x) is divided by x – c, then the remainder is ______.
f(c)
The Factor Theorem states that if f is a polynomial function and f(c) = 0, then _____ is a factor of f(x).
x – c
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
True or false
3/2 is a possible rational zero of f(x) = 2x^3 + 11x² – 7x – 6.
true
True or false
1/2 is a possible rational zero of f(x) = 3x^4 – 3x^3 + x² – x + 1
false
If a polynomial equation is of degree n, then counting multiple roots separately, the equation has _____ roots.
n
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
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
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
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
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
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
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, ∞)
y varies directly as x can be modeled by the equation _____, where k is called the _______.
y = kx
constant of variation
y varies directly as the nth power of x can be modeled by the equation _______.
y = kx^n
y varies inversely as x can be modeled by the equation _____.
y = k / x
y varies directly as x and inversely as z can be modeled by the equation _______.
y = kx / z
y varies jointly as x and z can be modeled by the equation ______.
y = kxz
In the equation S = 8A/P, S varies _____ as A and ______ as P.
directly
inversely
In the equation C = (0.02P1P2) / D², C varies _____ as P1 and P2 and _____ as the square of d.
jointly
inversely
