Which of the following polynomials could have the same end behavior as f(x)=ax^6+bx^5+c?
There may be more than one correct answer. Select all correct answers.
a. nx^5+bx^4+c
b. kx+b
c. dx^4−bx^3−cx^2+dx+e
d. −jx^8+bx^7+cx^4
e. −mx^2
can someone help me with this it is confusing and there is more than one answer so yea i need help
all polynomials of even degree either
go up on both ends
or do down on both ends,
depending on the sign of a.
Odd-degree polynomials go up on one end, and down on the other.
So, since f(x) is of degree 6 (even), any even-degree polynomial whose leading coefficient is the same sign as a will behave like f(x) at the ends.
So, C,D,E are possibilities. If all constants a,b,c,... are positive, then only C will do.
To determine which of the given polynomials could have the same end behavior as f(x) = ax^6 + bx^5 + c, let's examine the characteristics of the end behavior first.
For positive values of x, as x approaches infinity, the terms with the highest degrees in the polynomial will dominate, influencing the behavior of the function. Therefore, the end behavior of f(x) will be determined by the leading term, which is ax^6.
Since the coefficient a is nonzero, the end behavior of f(x) will be "up" on both ends, meaning that as x approaches positive or negative infinity, f(x) will tend to positive infinity.
Now, let's check each given polynomial and see if their end behavior matches that of f(x).
a. nx^5 + bx^4 + c: The highest degree term is nx^5. Here, the leading coefficient n is not necessarily the same as a, so the end behavior may not be the same as f(x). Therefore, this polynomial may not have the same end behavior as f(x).
b. kx + b: The highest degree term is kx, which is linear. As x approaches infinity, the linear function will also tend to infinity. However, since the degrees of the terms are different from f(x), this polynomial may not have the same end behavior.
c. dx^4 - bx^3 - cx^2 + dx + e: The highest degree term is dx^4. If the coefficient d is nonzero, then this polynomial will have the same end behavior as f(x), as x approaches infinity.
d. -jx^8 + bx^7 + cx^4: The highest degree term is -jx^8. Since the coefficient of the leading term is negative, the end behavior of this polynomial will be opposite to that of f(x), and therefore, the end behaviors do not match.
e. -mx^2: The highest degree term is -mx^2. Similar to the previous case, since the coefficient of the leading term is negative, the end behavior of this polynomial will be opposite to that of f(x), and they do not match.
In summary, the polynomials that could have the same end behavior as f(x) = ax^6 + bx^5 + c are:
c. dx^4 - bx^3 - cx^2 + dx + e
That's the only correct answer.
To determine which of the given polynomials could have the same end behavior as f(x) = ax^6 + bx^5 + c, we need to compare the leading terms of the polynomials.
The end behavior of a polynomial is determined by the sign (+/-) and the degree (power) of the leading term.
In the given polynomial f(x) = ax^6 + bx^5 + c, the leading term is ax^6.
Now let's compare the leading terms of the given answer choices to the leading term of f(x):
a. nx^5 + bx^4 + c - The leading term is nx^5. Since 5 is less than 6 (the degree of the leading term of f(x)), this polynomial does not have the same end behavior as f(x).
b. kx + b - The leading term is kx. Since 1 is less than 6, this polynomial does not have the same end behavior as f(x).
c. dx^4 - bx^3 - cx^2 + dx + e - The leading term is dx^4. Since 4 is less than 6, this polynomial does not have the same end behavior as f(x).
d. -jx^8 + bx^7 + cx^4 - The leading term is -jx^8. Since 8 is greater than 6, this polynomial does not have the same end behavior as f(x).
e. -mx^2 - The leading term is -mx^2. Since 2 is less than 6, this polynomial does not have the same end behavior as f(x).
From the given answer choices, none of the polynomials have the same end behavior as f(x) = ax^6 + bx^5 + c. Therefore, there is no correct answer in this case.