Math Problem Statement

(I) Identify the turning points, zeros, and x-intercepts. (ii) Do you find any point or zero which has a multiplicity in the graph? If so, specify them with multiplicity and explain the reason. (iii) Identify the degree and the polynomial as well as identify the domain in which the polynomial is increasing and decreasing. (iv) Do we have a local maximum/minimum here? If yes, find them. (v) Find the remainder when the polynomial is divided by x-4. vi) Write the functional equation of the graph.

Identify the turning points, zeros, x-intercepts, multiplicity of zeros, degree of the polynomial, local maxima/minima, remainder when divided by x-4, and the functional equation of the graph.

Solution

Let's analyze the graph step by step to address each part of the question.

(i) Identifying Turning Points, Zeros, and X-Intercepts:

  1. Turning Points: These are points where the graph changes direction (i.e., from increasing to decreasing or vice versa):

    • A: (1.659,3.124)(-1.659, -3.124)
    • B: (0.319,8.643)(0.319, 8.643)
  2. Zeros/X-Intercepts: These are points where the polynomial crosses the x-axis, where f(x)=0f(x) = 0:

    • D: (2,0)(-2, 0)
    • E: (1,0)(-1, 0)
    • C: (2,0)(2, 0)

Thus, the x-intercepts are x=2,1,2x = -2, -1, 2.

(ii) Multiplicity of Zeros:

Multiplicity refers to the number of times a particular zero appears in the factored form of the polynomial. If the graph touches the x-axis and bounces off, the zero has even multiplicity. If it crosses the axis, it has odd multiplicity.

  • At E (1,0)(-1, 0), the graph touches and bounces back, suggesting this zero has even multiplicity, likely a multiplicity of 2.
  • The other intercepts at D (2,0)(-2, 0) and C (2,0)(2, 0) show the graph crosses the x-axis, suggesting these zeros have an odd multiplicity, likely multiplicity of 1.

(iii) Degree of Polynomial and Behavior:

  • Based on the turning points and x-intercepts, the polynomial seems to be a degree 4 polynomial. It has 3 distinct real zeros (though one is of multiplicity 2).
  • The general behavior of the graph suggests a quartic polynomial: the graph starts negative, rises to a maximum, drops to a minimum, then rises again.

Increasing and Decreasing Intervals:

  • Increasing on (1.659,0.319)(-1.659, 0.319) and (2,)(2, \infty)
  • Decreasing on (,1.659)(-\infty, -1.659), (0.319,2)(0.319, 2)

(iv) Local Maximum and Minimum:

  • Local Maximum: At B (0.319,8.643)(0.319, 8.643)
  • Local Minimum: At A (1.659,3.124)(-1.659, -3.124)

(v) Remainder when Dividing by x4x-4:

The remainder theorem states that when a polynomial f(x)f(x) is divided by xax-a, the remainder is f(a)f(a). Here, we want the remainder when divided by x4x - 4, so we need to find f(4)f(4). To do this accurately, we would need the functional form of the polynomial, but the image doesn't provide the function explicitly. You would need to calculate f(4)f(4) using the function if it's given or by constructing it using the points from the graph.

(vi) Functional Equation of the Graph:

The general form of the polynomial is likely of the form: f(x)=a(x+2)(x+1)2(x2)f(x) = a(x + 2)(x + 1)^2(x - 2) where:

  • The zero at x=2x = -2 has multiplicity 1,
  • The zero at x=1x = -1 has multiplicity 2 (since the graph bounces off),
  • The zero at x=2x = 2 has multiplicity 1.

You would need to determine the leading coefficient aa using another point on the graph, such as the local maximum or minimum. For example, using point B(0.319,8.643)B(0.319, 8.643), you could solve for aa.


Follow-Up Questions:

  1. How can the behavior at the x-axis determine the multiplicity of a zero in a polynomial?
  2. Why is the degree of the polynomial important for determining end behavior?
  3. How would the graph look different if the zero at x=1x = -1 had a higher multiplicity than 2?
  4. Can you explain how we can find the leading coefficient aa using the known points on the graph?
  5. What impact does the remainder theorem have when dividing polynomials?

Tip: For any polynomial graph, always check the behavior at turning points and x-intercepts to determine multiplicity and degree.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Graph Analysis
Turning Points
Zeros and X-Intercepts
Multiplicity of Zeros
Local Maxima and Minima

Formulas

Polynomial equation f(x)
Multiplicity of zeros
Remainder Theorem

Theorems

Remainder Theorem
Multiplicity Theorem
Fundamental Theorem of Algebra

Suitable Grade Level

Grades 11-12