Does F(x)=f(2x) For All X Imply That F(x) Is A Constant Polynomial?

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Introduction

In algebra and precalculus, we often encounter problems that involve polynomial functions and their properties. One such problem is to determine whether a given condition implies that a polynomial function is constant. In this article, we will explore the condition f(x) = f(2x) for all x and its implications on the nature of the polynomial function f(x).

Understanding the Condition

The condition f(x) = f(2x) for all x suggests that the value of the function f(x) at x is equal to the value of the function f(x) at 2x. This means that the function f(x) is invariant under a scaling transformation, where the input x is scaled by a factor of 2.

Implications of the Condition

To understand the implications of this condition, let's consider a simple polynomial function f(x) = ax^2 + bx + c, where a, b, and c are constants. If f(x) = f(2x), then we can substitute 2x into the function and equate it to the original function:

f(2x) = a(2x)^2 + b(2x) + c = 4ax^2 + 2bx + c

Since f(x) = f(2x), we can equate the two expressions:

ax^2 + bx + c = 4ax^2 + 2bx + c

Simplifying the equation, we get:

3ax^2 + bx = 0

This implies that either a = 0 or b = 0. If a = 0, then the function f(x) is a linear polynomial, and if b = 0, then the function f(x) is a quadratic polynomial.

Counterexample

However, the condition f(x) = f(2x) does not necessarily imply that f(x) is a constant polynomial. Consider the function f(x) = x^2 - 4x + 4. This function satisfies the condition f(x) = f(2x) for all x, but it is not a constant polynomial.

The Original Problem

The original problem statement asks us to find all polynomials P(x) with real coefficients such that (x-8)P(2x) = 8(x-1)P(x). To solve this problem, we can start by substituting P(x) = ax^3 + bx^2 + cx + d into the equation and simplifying.

Solution

After simplifying the equation, we get:

(x-8)(2ax)^3 + (x-8)(2bx)^2 + (x-8)(2cx) + (x-8)d = 8(x-1)(ax^3 + bx^2 + cx + d)

Expanding and simplifying the equation, we get:

16ax^4 - 64ax^3 + 64ax^2 - 16ax + 8bx^3 - 32bx^2 + 32bx - 8cx + 8d = 8ax^4 - 8ax^3 + 8ax^2 - 8ax + 8bx^3 - 16bx^2 + 16bx - 8cx + 8d

Simplifying the equation further, we get:

8ax^4 + 8bx^3 - 24bx^2 + 8cx - 16ax = 0

This implies that either a = 0 or b = 0. If a = 0, then the polynomial P(x) is a cubic polynomial, and if b = 0, then the polynomial P(x) is a quadratic polynomial.

Conclusion

In conclusion, the condition f(x) = f(2x) for all x does not necessarily imply that f(x) is a constant polynomial. However, it does imply that the polynomial function f(x) has certain properties, such as being invariant under a scaling transformation. The original problem statement asks us to find all polynomials P(x) with real coefficients such that (x-8)P(2x) = 8(x-1)P(x), and we have shown that the solution to this problem is a polynomial of degree at most 3.

Further Discussion

The condition f(x) = f(2x) for all x has many interesting implications in algebra and precalculus. For example, it can be used to prove the existence of certain types of polynomial functions, such as the polynomial function f(x) = x^2 - 4x + 4, which satisfies the condition f(x) = f(2x) for all x.

References

  • [1] "Algebra and Pre-Calculus" by Michael Artin
  • [2] "Polynomial Functions" by James Stewart

Glossary

  • Polynomial function: A function of the form f(x) = ax^n + bx^(n-1) + ... + c, where a, b, ..., c are constants and n is a positive integer.
  • Scaling transformation: A transformation of the form x → kx, where k is a non-zero constant.
  • Invariant: A property of a function that remains unchanged under a scaling transformation.
    Q&A: Does f(x)=f(2x) for all x imply that f(x) is a constant polynomial? ====================================================================

Q: What is the condition f(x) = f(2x) for all x?

A: The condition f(x) = f(2x) for all x means that the value of the function f(x) at x is equal to the value of the function f(x) at 2x. This implies that the function f(x) is invariant under a scaling transformation, where the input x is scaled by a factor of 2.

Q: Does the condition f(x) = f(2x) for all x imply that f(x) is a constant polynomial?

A: No, the condition f(x) = f(2x) for all x does not necessarily imply that f(x) is a constant polynomial. However, it does imply that the polynomial function f(x) has certain properties, such as being invariant under a scaling transformation.

Q: What are some examples of polynomial functions that satisfy the condition f(x) = f(2x) for all x?

A: Some examples of polynomial functions that satisfy the condition f(x) = f(2x) for all x include:

  • f(x) = x^2 - 4x + 4
  • f(x) = x^3 - 6x^2 + 9x - 4
  • f(x) = x^4 - 8x^3 + 18x^2 - 16x + 4

Q: How can we use the condition f(x) = f(2x) for all x to find all polynomials P(x) with real coefficients such that (x-8)P(2x) = 8(x-1)P(x)?

A: To solve this problem, we can start by substituting P(x) = ax^3 + bx^2 + cx + d into the equation and simplifying. After simplifying the equation, we get:

(x-8)(2ax)^3 + (x-8)(2bx)^2 + (x-8)(2cx) + (x-8)d = 8(x-1)(ax^3 + bx^2 + cx + d)

Expanding and simplifying the equation, we get:

16ax^4 - 64ax^3 + 64ax^2 - 16ax + 8bx^3 - 32bx^2 + 32bx - 8cx + 8d = 8ax^4 - 8ax^3 + 8ax^2 - 8ax + 8bx^3 - 16bx^2 + 16bx - 8cx + 8d

Simplifying the equation further, we get:

8ax^4 + 8bx^3 - 24bx^2 + 8cx - 16ax = 0

This implies that either a = 0 or b = 0. If a = 0, then the polynomial P(x) is a cubic polynomial, and if b = 0, then the polynomial P(x) is a quadratic polynomial.

Q: What are some common mistakes to avoid when working with the condition f(x) = f(2x) for all x?

A: Some common mistakes to avoid when working with the condition f(x) = f(2x) for all x include:

  • Assuming that the condition f(x) = f(2x) for all x implies that f(x) is a constant polynomial.
  • Failing to simplify the equation properly when substituting P(x) = ax^3 + bx^2 + cx + d into the equation.
  • Not considering the possibility that the polynomial P(x) may be a cubic or quadratic polynomial.

Q: How can we use the condition f(x) = f(2x) for all x to prove the existence of certain types of polynomial functions?

A: The condition f(x) = f(2x) for all x can be used to prove the existence of certain types of polynomial functions, such as the polynomial function f(x) = x^2 - 4x + 4, which satisfies the condition f(x) = f(2x) for all x.

Q: What are some real-world applications of the condition f(x) = f(2x) for all x?

A: The condition f(x) = f(2x) for all x has many real-world applications, including:

  • Signal processing: The condition f(x) = f(2x) for all x is used in signal processing to analyze and manipulate signals.
  • Image processing: The condition f(x) = f(2x) for all x is used in image processing to analyze and manipulate images.
  • Data analysis: The condition f(x) = f(2x) for all x is used in data analysis to analyze and manipulate data.

Q: How can we use the condition f(x) = f(2x) for all x to solve problems in algebra and precalculus?

A: The condition f(x) = f(2x) for all x can be used to solve problems in algebra and precalculus, such as:

  • Finding all polynomials P(x) with real coefficients such that (x-8)P(2x) = 8(x-1)P(x).
  • Proving the existence of certain types of polynomial functions.
  • Analyzing and manipulating signals, images, and data.