Math Problem Statement

solve

The following graphs represent polynomial functions of 4th degree. Which of these graphs represents the 4th degree polynomial function with two distinct real zeros and two complex ones?

Solution

To solve this problem, let's analyze the requirements and understand the characteristics of the polynomial graph we're looking for.

Given:

  1. The polynomial is of the 4th degree, which means its general form is f(x)=ax4+bx3+cx2+dx+ef(x) = ax^4 + bx^3 + cx^2 + dx + e.
  2. We need to identify a graph that represents a 4th degree polynomial with:
    • Two distinct real zeros: This implies that the polynomial will cross the x-axis at two distinct points.
    • Two complex zeros: Complex zeros occur in conjugate pairs and do not result in x-axis intersections.

Characteristics of the Desired Graph:

  1. End Behavior: Since it's a 4th degree polynomial, the ends of the graph will either both go upwards (if the leading coefficient aa is positive) or both go downwards (if aa is negative).
  2. Intercepts with the x-axis: We expect exactly two points where the graph intersects the x-axis because the polynomial has two real roots.
  3. Behavior around Zeros: The graph should only cross the x-axis at two points, as the remaining two roots are complex (and hence do not intersect the x-axis).

Steps:

  1. Look at each graph for the one that crosses the x-axis exactly twice (indicating two real roots).
  2. Ensure that the graph has the proper end behavior consistent with a 4th degree polynomial (both ends of the graph should point in the same direction).

Please refer to the uploaded graphs and identify the one that matches these criteria. Let me know if you need further assistance with choosing the specific graph.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Functions
Complex Numbers
Zeros of Polynomials

Formulas

Polynomial function of the form f(x) = ax^4 + bx^3 + cx^2 + dx + e

Theorems

Fundamental Theorem of Algebra
Behavior of Polynomial Roots

Suitable Grade Level

Grades 10-12