교과서 질문
Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=3x4-7x3-6x2+12x+8
678
views
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
Not the one you use?Change textbookSelect a different textbook
Table of contents
4장, 문제 45
If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a) domain (b) range (c) end behavior (d) number of zeros (e) number of turning points 10x7
검증된 단계별 안내1
Identify the dominating term of the polynomial function, which is given as \$10x^{7}\(. This term determines the overall behavior of the polynomial for very large or very small values of \)x$.
For (a) domain: Recall that polynomial functions are defined for all real numbers, so the domain is all real numbers, \((-\infty, \infty)\), regardless of the dominating term.
For (b) range: Since the dominating term is \$10x^{7}\(, an odd-degree term with a positive leading coefficient, the range will be all real numbers, \((-\infty, \infty)\), because the function will take on very large positive and negative values as \)x$ becomes large in magnitude.
For (c) end behavior: Analyze the dominating term \$10x^{7}$. Because the degree is odd and the leading coefficient is positive, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\). This describes the end behavior of the polynomial.
For (d) number of zeros and (e) number of turning points: The maximum number of real zeros is equal to the degree of the polynomial, which is 7. The maximum number of turning points is one less than the degree, so up to 6 turning points. The dominating term sets these maximum possible values, but the actual number depends on the specific polynomial.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
9m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Dominating Term of a Polynomial
The dominating term of a polynomial is the term with the highest degree, which determines the overall shape and behavior of the graph for large values of x. In this case, 10x⁷ is the dominating term, meaning the polynomial behaves like 10x⁷ as x approaches infinity or negative infinity.
추천 영상:
가이드 코스
05:13
Introduction to Polynomials
Domain and Range of Polynomial Functions
The domain of any polynomial function is all real numbers since polynomials are defined everywhere. The range depends on the degree and leading coefficient; for odd-degree polynomials like x⁷, the range is all real numbers because the graph extends infinitely in both positive and negative directions.
추천 영상:
4:22Domain & Range of Transformed Functions
End Behavior, Zeros, and Turning Points
The end behavior of a polynomial is determined by the leading term: for 10x⁷, as x → ∞, f(x) → ∞, and as x → -∞, f(x) → -∞. The maximum number of zeros is equal to the degree (7), and the maximum number of turning points is one less than the degree (6), reflecting the possible changes in direction of the graph.
추천 영상:
06:08End Behavior of Polynomial Functions
관련 실천
교과서 질문
Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2+1)/(x2+9)
1154
views
교과서 질문
Period of a Pendulum The period of a pendulum varies directly as the square root of the length of the pendulum and inversely as the square root of the acceleration due to gravity. Find the period when the length is 121 cm and the acceleration due to gravity is 980 cm per second squared, if the period is 6π seconds when the length is 289 cm and the acceleration due to gravity is 980 cm per second squared.
561
views
교과서 질문
Several graphs of the quadratic function ƒ(x) = ax2 + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) a < 0; b2 - 4ac = 0
1072
views
교과서 질문
Solve each polynomial inequality. Give the solution set in interval notation. x5 + x2 + 2 ≥ x4 + x3 + 2x
489
views
교과서 질문
For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 6x4 + x3 - 8x2 + 5x+6; k=1/2
501
views