Math Problem Statement

Given the polynomial function \( f(x) = 2x^3 + 3x^2 - 2x - 3 \), which of the following statements are true?

Solution

To address the question, let's analyze the given polynomial function f(x)=2x3+3x22x3f(x) = 2x^3 + 3x^2 - 2x - 3 and evaluate each statement to determine its truthfulness:

  1. Statement: "As x+x \to +\infty, y+y \to +\infty"

    • Since f(x)f(x) is a cubic polynomial with a positive leading coefficient (22), as x+x \to +\infty, the behavior of f(x)f(x) will follow the leading term 2x32x^3. Therefore, f(x)+f(x) \to +\infty as x+x \to +\infty.
    • This statement is true.
  2. Statement: " x=3x = -3 is a zero."

    • To check if x=3x = -3 is a zero, substitute x=3x = -3 into f(x)f(x): f(3)=2(3)3+3(3)22(3)3=54+27+63=24f(-3) = 2(-3)^3 + 3(-3)^2 - 2(-3) - 3 = -54 + 27 + 6 - 3 = -24
    • Since f(3)0f(-3) \neq 0, x=3x = -3 is not a zero.
    • This statement is false.
  3. Statement: "It has the same end behavior as f(x)=x5f(x) = x^5."

    • The function f(x)=2x3+3x22x3f(x) = 2x^3 + 3x^2 - 2x - 3 is a cubic function, so its end behavior will match that of f(x)=x3f(x) = x^3, not x5x^5. Cubic and quintic functions have different end behaviors because cubic functions have opposite behaviors at ++\infty and -\infty, while quintic functions (odd-powered, leading coefficient positive) have similar behaviors.
    • This statement is false.
  4. Statement: " (x2)(x - 2) is a factor."

    • To determine if (x2)(x - 2) is a factor, apply the Factor Theorem by substituting x=2x = 2 into f(x)f(x): f(2)=2(2)3+3(2)22(2)3=16+1243=21f(2) = 2(2)^3 + 3(2)^2 - 2(2) - 3 = 16 + 12 - 4 - 3 = 21
    • Since f(2)0f(2) \neq 0, (x2)(x - 2) is not a factor.
    • This statement is false.
  5. Statement: "As x+x \to +\infty, yy \to -\infty"

    • As discussed in statement 1, the polynomial's leading term is 2x32x^3, which means f(x)+f(x) \to +\infty as x+x \to +\infty, not -\infty.
    • This statement is false.
  6. Statement: " x=32x = -\frac{3}{2} is a zero."

    • To check if x=32x = -\frac{3}{2} is a zero, substitute x=32x = -\frac{3}{2} into f(x)f(x): f(32)=2(32)3+3(32)22(32)3f\left(-\frac{3}{2}\right) = 2\left(-\frac{3}{2}\right)^3 + 3\left(-\frac{3}{2}\right)^2 - 2\left(-\frac{3}{2}\right) - 3 Simplifying this calculation shows f(32)=0f\left(-\frac{3}{2}\right) = 0.
    • This statement is true.

Conclusion:

The true statements are:

  1. "As x+x \to +\infty, y+y \to +\infty"
  2. " x=32x = -\frac{3}{2} is a zero."

Would you like further details on any specific step?

Here are some related questions to expand your understanding:

  1. How can we determine end behavior of polynomials based on their degree and leading coefficient?
  2. What are the methods to verify if a term is a factor of a polynomial?
  3. Why does a polynomial’s degree impact its end behavior?
  4. How can the Factor Theorem be applied to find polynomial roots?
  5. 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 xx.

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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