Math Problem Statement
Solution
To address the question, let's analyze the given polynomial function and evaluate each statement to determine its truthfulness:
-
Statement: "As , "
- Since is a cubic polynomial with a positive leading coefficient (), as , the behavior of will follow the leading term . Therefore, as .
- This statement is true.
-
Statement: " is a zero."
- To check if is a zero, substitute into :
- Since , is not a zero.
- This statement is false.
-
Statement: "It has the same end behavior as ."
- The function is a cubic function, so its end behavior will match that of , not . Cubic and quintic functions have different end behaviors because cubic functions have opposite behaviors at and , while quintic functions (odd-powered, leading coefficient positive) have similar behaviors.
- This statement is false.
-
Statement: " is a factor."
- To determine if is a factor, apply the Factor Theorem by substituting into :
- Since , is not a factor.
- This statement is false.
-
Statement: "As , "
- As discussed in statement 1, the polynomial's leading term is , which means as , not .
- This statement is false.
-
Statement: " is a zero."
- To check if is a zero, substitute into : Simplifying this calculation shows .
- This statement is true.
Conclusion:
The true statements are:
- "As , "
- " is a zero."
Would you like further details on any specific step?
Here are some related questions to expand your understanding:
- How can we determine end behavior of polynomials based on their degree and leading coefficient?
- What are the methods to verify if a term is a factor of a polynomial?
- Why does a polynomial’s degree impact its end behavior?
- How can the Factor Theorem be applied to find polynomial roots?
- What techniques can simplify calculations when substituting values in complex polynomials?
Tip: When evaluating end behavior, focus only on the term with the highest degree as it dominates the function's growth at extreme values of .
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
End Behavior of Polynomials
Factor Theorem
Formulas
End behavior is determined by the leading term of a polynomial
Factor Theorem: If \(f(a) = 0\), then \(x - a\) is a factor of \(f(x)\)
Theorems
Factor Theorem
Suitable Grade Level
Grades 10-12
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