19 20 Polynomials Flashcards
What is a polynomial?
An expression in the form:
a0 + a1x + a2x² + … + anxn
(listed in reverse order in its standard form)
a is just any number
all exponents are positive integers
Can √5 be a coefficient?
An exponent?
√5 can be a coefficient.
√5 cannot be an exponent (it is not an integer)
Can -2 be a coefficient?
An exponent?
- 2 can be a coefficient
- 2 cannot be an exponent because it is not a positive integer
Can √x be in a polynomial?
No.
Because √x = x½ and exponents must be positive integers
Can (x² + 2x)/(x³ + 1) be a polynomial?
No.
The x³ in the denominator is x⅓ which is not a positive integer.
What is the degree of a polynomial?
What are the names of the first 5 polynomial degrees?
The highest exponent.
linear
quadratic
cubic
quartic
quintic
What is the leading coefficient of a polynomial?
The coefficient of the highest degree polynomial
How do you classify polynomials by quantity of terms?
monomial → 3x²
binomial → 3x² + 2
trinomial → 3x² + 2x + 1
Classify 5x³ -1
a cubic binomial
Classify A(x) = πx²
a quadratic monomial
What is the domain of all polynomials?
AR#s
What does the range of polynomials depend on?
The degree.
If odd, the range is AR#s.
If even, the range is not AR#s because both ends will either go up or down so the range is restricted.
Can the graph of a polynomial have a sharp corner?
No. That’s why x⅔ is not a polynomial.
It has a sharp corner at the origin.
Can the graph of a polynomial have a break?
No. Because the domain is AR#s.
What is the maximum number of x-intercepts a polynomial can have?
the degree of the polynomial
What happens to the ends of the graph of a polynomial?
They never flatten out.
If it is an even polynomial, they point in the same direction.
If odd, they point in opposite directions.
In an even coefficient, how do you know if the ends point up or down?
If the lead coefficient is positive, they point up.
This is because negative xs give the same y as the corresponding positive x
In an odd coefficient, how do you know if which end points up?
If the lead coefficient is positive, the left side points down and the right side points up.
In other words, the graph is always going up
Why does the lead coefficient control the direction of the ends?
It is the dominant term (has greatest effect)
For f(x) = x³ :
classify
how many x-intercepts?
ends up or down?
It is a cubic monomial.
1 x-intercept (0,0)
points down on the left, up on the right
How many turns does the graph of a cubic polynomial have?
A cubic polynomial can have, at most, 3 roots, so only 2 turns.
Cubic polynomials have either 0 or 2 turns
What is the x-intercept of
f(x) = x³ + 4
(0,4)
How does the graph of
x³ - 3x² + 3x -1
compare to
x³ ?
x³ - 3x² + 3x -1 =
(x - 1)³
This is the graph of x³ moved 1 to the right.
How many x-intercepts are there in:
f(x) = x4 - 9x²
Which direction do the ends go?
f(x) = x4 - 9x²
= x²(x² - 9)
= x²(x - 3)(x + 3)
3 roots: x = 0 and ±3
both ends point in the same direction (even exponent) and they go up (positive coefficient)
What is the end behavior of
f(x) = 3x6 - 2x + 7
degree is even
leading exponent is positive
Both ends point up
What is the end behavior of
f(x) = -4x5 + 3x4 + 2x²
degree = odd
lead coefficient = negative
left side points up
right side points down
What is the end behavior of
f(x) = -10x8 + 20
degree = even
lead coefficient = negative
Both ends point down
What is the end behavior of
f(x) = (x² - 1)²
degree = even
lead coefficient = positive
Both ends point up
