Math Problem Statement

For each function, determine whether it is a polynomial function.

Solution

Let's examine each function to determine whether it is a polynomial.

(a) h(x)=3x(x+1)(x4)2h(x) = 3x(x + 1)(x - 4)^2

This is a product of polynomials, which is expanded as a polynomial. Polynomials are defined as expressions with non-negative integer powers of xx, and this meets that requirement. Answer: Yes, this is a polynomial.

(b) u(x)=14x8x8+4u(x) = \frac{1}{4x} - 8x^8 + 4

This function contains the term 1x\frac{1}{x}, which has a negative power of xx (i.e., x1x^{-1}). A polynomial function cannot have negative exponents. Answer: No, this is not a polynomial.

(c) f(x)=4x3+6x7f(x) = -4x^3 + 6x^7

This function is a sum of terms with positive integer exponents of xx. It fits the definition of a polynomial. Answer: Yes, this is a polynomial.

(d) v(x)=6xv(x) = 6\sqrt{x}

This function contains a square root of xx, which is equivalent to x1/2x^{1/2}. Since polynomial exponents must be non-negative integers, this is not a polynomial. Answer: No, this is not a polynomial.

Would you like further details on any of these functions or have additional questions?

Related Questions:

  1. What are the specific conditions for a function to be classified as a polynomial?
  2. How do negative exponents affect the classification of a polynomial?
  3. What is the process for expanding h(x)=3x(x+1)(x4)2h(x) = 3x(x + 1)(x - 4)^2 into standard polynomial form?
  4. Why does the presence of square roots or fractional powers exclude a function from being a polynomial?
  5. What are common misconceptions when identifying polynomials?

Tip:

Remember, a polynomial function's terms must all have non-negative integer exponents.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Exponents
Square Roots

Formulas

Polynomial function: P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_0
Exponent rule: x^(m/n) = n√(x^m)

Theorems

Polynomials cannot have negative or fractional exponents
A product of polynomials is a polynomial

Suitable Grade Level

Grades 9-12