Math Problem Statement

List all real zeros of f. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

Solution

To list all real zeros of a function f(x)f(x), you would first need to:

  1. Analyze the equation: I would need the specific form of the function f(x)f(x) to determine its zeros.
  2. Set the function equal to zero: Find where f(x)=0f(x) = 0, since zeros are the points where the function crosses or touches the x-axis.
  3. Solve the equation: Use appropriate algebraic techniques (factoring, quadratic formula, etc.) to solve for xx.
  4. Include repetitions: If any of the roots appear multiple times (as repeated factors), you should list them with the correct multiplicity.

Please provide the explicit form of the function f(x)f(x), and I will help you find the real zeros.


Would you like help with one of these next steps?

  1. How to identify multiplicity in repeated zeros?
  2. Techniques for factoring higher-degree polynomials?
  3. Understanding when functions might not have real zeros.
  4. Using the quadratic formula to find zeros.
  5. Graphical methods to approximate or visualize real zeros.

Tip: Repeated zeros occur when the factor of (xa)n(x - a)^n appears in the function, where n>1n > 1.

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Math Problem Analysis

Mathematical Concepts

Algebra
Real Zeros
Roots of Functions
Factoring
Multiplicity

Formulas

f(x) = 0
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Factoring: f(x) = (x - a)^n

Theorems

Fundamental Theorem of Algebra
Multiplicity of Roots

Suitable Grade Level

Grades 9-12