Problem 73 - Algebra and Trigonometry

Question

Recall that a function \(f\) is odd if \(f(-x)=-f(x)\) or even if \(f(-x)=f(x)\) for all real \(x .\) $$\begin{array}{l}{\text { (a) Show that a polynomial } P(x) \text { that contains only odd }} \\ {\text { powers of } x \text { is an odd function. }} \\\ {\text { (b) Show that a polynomial } P(x) \text { that contains only even }} \\ {\text { powers of } x \text { is an even function. }} \\ {\text { (c) Show that if a polynomial } P(x) \text { contains both odd and }} \\ {\text { even powers of } x, \text { then it is neither an odd nor an even }} \\\ {\text { function. }}\end{array}$$ $$ \begin{array}{l}{\text { (d) Express the function }} \\ {\qquad P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5} \\ {\text { as the sum of an odd function and an even function. }}\end{array} $$

Step-by-Step Solution

Answer

(a) Odd; (b) Even; (c) Neither odd nor even; (d) \( P(x) = (x^5 + 6x^3 - 2x) + (-x^2 + 5) \).

Step 1: Definition of Odd and Even Functions

A function is called odd if \( f(-x) = -f(x) \) for all \( x \), and it is called even if \( f(-x) = f(x) \) for all \( x \). We will use these definitions to show properties of polynomials containing odd and even powers of \( x \).

Step 2: Prove a Polynomial with Only Odd Powers is an Odd Function

Consider a polynomial \( P(x) \) with only odd powers: \( P(x) = a_n x^n + a_{n-2} x^{n-2} + \, \ldots \, + a_1 x^1 \), where all powers are odd. We evaluate \( P(-x) \):\[ P(-x) = a_n (-x)^n + a_{n-2} (-x)^{n-2} + \, \ldots \, + a_1 (-x)^1 = -a_n x^n - a_{n-2} x^{n-2} - \, \ldots \, - a_1 x^1 = -P(x) \]Thus, \( P(x) \) is an odd function.

Step 3: Prove a Polynomial with Only Even Powers is an Even Function

Consider a polynomial \( P(x) \) with only even powers: \( P(x) = a_m x^m + a_{m-2} x^{m-2} + \, \ldots \, + a_0 x^0 \), where all powers are even. We evaluate \( P(-x) \):\[ P(-x) = a_m (-x)^m + a_{m-2} (-x)^{m-2} + \, \ldots \, + a_0 = a_m x^m + a_{m-2} x^{m-2} + \, \ldots \, + a_0 = P(x) \]Thus, \( P(x) \) is an even function.

Step 4: Show a Polynomial with Both Odd and Even Powers is Neither Odd nor Even

Consider a polynomial \( P(x) \) with both odd and even powers, e.g., \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \, \ldots \, + a_1 x + a_0 \). Since both odd and even terms are present, \( P(-x) \equiv a_n (-x)^n + \, \ldots \, + a_0 eq P(x) \) and \( P(-x) eq -P(x) \).The presence of both odd and even powers means \( P(x) \) satisfies neither definition strictly and is neither odd nor even.

Step 5: Expressing \( P(x) = x^5 + 6x^3 - x^2 - 2x + 5 \) as Sum of Odd and Even Functions

Write \( P(x) \) separating odd and even power terms:Odd function: \( P_{odd}(x) = x^5 + 6x^3 - 2x \)Even function: \( P_{even}(x) = -x^2 + 5 \)Therefore, the expression for \( P(x) \) as a sum of an odd and even function is:\[ P(x) = (x^5 + 6x^3 - 2x) + (-x^2 + 5) \]

Step 6: Verify that \( P_{odd}(x) \) and \( P_{even}(x) \) Have Desired Properties

Check \( P_{odd}(-x) = -x^5 - 6x^3 + 2x = -P_{odd}(x) \); it is odd.Check \( P_{even}(-x) = -x^2 + 5 = P_{even}(x) \); it is even.

Key Concepts

Odd and Even Functions

Understanding whether a function is odd or even is essential for classifying polynomial functions. A function is called **odd** if it satisfies the condition \( f(-x) = -f(x) \) for all real numbers \( x \). An **even** function, on the other hand, meets the condition \( f(-x) = f(x) \).

In polynomials, these concepts are linked to the powers of \( x \). If a polynomial contains only odd powers of \( x \), such as \( x^1, x^3, x^5, \) and so on, we can demonstrate that it is an odd function. This is because when we substitute \(-x\) for \(x\), each term also changes sign, leading to \(-P(x)\). Similarly, a polynomial with only even powers of \( x \) is an even function. In this case, substituting \(-x\) does not change the sign of any term, resulting in the same polynomial, \(P(x)\).

Polynomials with a mix of odd and even powers are neither completely matched by \( f(-x) = -f(x) \) nor \( f(-x) = f(x) \), so they are classified as neither odd nor even. By dividing a mixed polynomial into its odd and even parts, one can express it as a combination of an odd and an even function, showcasing an important property of polynomial functions.

Algebraic Expressions

Algebraic expressions are foundational elements of algebra involving combinations of numbers, variables, and arithmetic operations. They are crucial for constructing and understanding polynomial functions.

A polynomial is a type of algebraic expression that consists of terms which are sums or differences of variables raised to whole number powers. For instance, \( x^5 + 6x^3 - x^2 - 2x + 5 \) is a polynomial. It systematically combines terms like these, involving coefficients (numbers in front of variable powers) and variables themselves.

• The term \( x^5 \) is an example where \( x \) is raised to an odd power.
• The term \(-x^2\), in contrast, demonstrates \( x \) raised to an even power.
• Constant terms like \( 5 \) can also be thought of as \( x^0 \), which is an even power of \( x \).

When constructing polynomials from algebraic expressions, understanding the role of these terms and how they contribute to the function's behavior is crucial. By arranging and rearranging these components, one can explore properties and classify the function as odd, even, or neither.

Polynomial Properties

Polynomials exhibit a variety of properties that significantly influence their behavior and classification. These properties are essential for understanding and working with polynomial functions in algebra.

**Degree of a Polynomial:**The degree of a polynomial is defined by the highest power of \( x \) present in the expression. For example, the degree of \( x^5 + 6x^3 - x^2 - 2x + 5 \) is 5.

• The degree helps in predicting the end behavior of the polynomial function—that is, how the function behaves as the input values become very large or very small.

**Symmetry of Polynomial Functions:**Polynomials can be symmetric about the y-axis (even functions) or the origin (odd functions), depending on the powers of \( x \) involved.

• Even polynomials, with all even powers, show symmetry about the y-axis.
• Odd polynomials, with all odd powers, show symmetry with respect to the origin.

Combining both even and odd terms in the polynomial leads to a function that typically lacks symmetry, making it neither purely odd nor purely even. Recognizing these properties enhances our understanding of how different parts of the polynomial influence its overall graph and behavior.