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Chapter 2: Problem 10

Determine which functions are polynomials, and for those that are, state their degree. $$H(x)=\frac{x^{2}+1}{2}$$

Short Answer

Expert verified
The function is a polynomial of degree 2.

Step by step solution

01

Identify the Form

First, identify if the given function, \( H(x) = \frac{x^2 + 1}{2} \), is in the form of a polynomial. A polynomial function is a mathematical expression involving a sum of powers of \( x \) with non-negative integer exponents and real coefficients. It must not appear as a ratio of polynomials unless it can be simplified to a polynomial.
02

Simplify the Expression

Simplify \( H(x) = \frac{x^2 + 1}{2} \) to check if division results in a simple polynomial. Divide each term in the numerator by 2: \( H(x) = \frac{1}{2}x^2 + \frac{1}{2}. \) This form shows that each term is indeed a multiple of \( x^2 \) with non-negative powers.
03

Verify Polynomial Terms

Check that all terms \( \frac{1}{2}x^2 \) and \( \frac{1}{2} \) consist only of non-negative integer powers of \( x \). The term \( \frac{1}{2} \) corresponds to the constant term with a power of \( x^0 \). Both terms satisfy the definition of a polynomial.
04

Determine the Degree

The degree of a polynomial is the highest power of \( x \) that occurs with a non-zero coefficient. In the expression \( \frac{1}{2}x^2 + \frac{1}{2} \), the highest power of \( x \) is 2, which makes this a polynomial of degree 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of a Polynomial
In mathematics, the degree of a polynomial is a crucial concept as it gives insight into the polynomial's behavior and graph. The degree is simply the highest power of the variable in the polynomial with a non-zero coefficient. Take for example the polynomial expression \( \frac{1}{2}x^2 + \frac{1}{2} \). Here we can see that the term with the highest power of \( x \) is \( \frac{1}{2}x^2 \). Hence, the degree of this polynomial is 2.
  • The degree determines the shape and potential intersections of the graph.
  • A polynomial of degree 2 is known as a quadratic polynomial and it usually forms a parabolic shape graph.
  • Recognizing the degree helps in solving polynomial equations and understanding the curve's steepness and direction.
Always ensure you look for the highest power when determining the degree, as this dictates the leading behavior of the polynomial as \( x \) approaches large positive or negative values.
Simplification of Expressions
Simplifying algebraic expressions is a key skill in mathematics that allows expressions to be more easily understood and manipulated. Sometimes polynomial expressions are presented in a complex form such as a fraction. The given function \( H(x) = \frac{x^2 + 1}{2} \) is an example of this: the expression appears as a fractional division.
  • To simplify, check if the fraction can be expressed as a simpler polynomial.
  • Divide each term in the numerator separately by the denominator: \( H(x) = \frac{1}{2}x^2 + \frac{1}{2} \).
  • This results in a clear linear combination of terms that represent a polynomial.
Simplifying expressions reduces complexity and makes identifying polynomial properties easier. It's especially helpful in solving equations where a simplified form provides clearer insight into the solution and behavior of the expression.
Polynomial Terms
Polynomial functions are made up of terms. Each term is composed of a coefficient, the variable (usually \( x \)), and the power to which the variable is raised. Understanding these terms helps to see what makes up a polynomial. Consider the polynomial \( \frac{1}{2}x^2 + \frac{1}{2} \).
  • Each term has a distinct role and characteristic:
  • \( \frac{1}{2}x^2 \) is the term with degree 2, contributing to the quadratic nature of the polynomial.
  • \( \frac{1}{2} \) is the constant term, unaffected by the variable, considered as having degree 0 (since \( x^0 = 1 \)).
  • Terms with negative powers or variables in the denominator would disqualify an expression from being a polynomial.
Understanding each term allows a better grasp of the overall structure and behavior, crucial for graphing, evaluating, and manipulating polynomial functions.

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