Problem 77 - 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 \(x\) . (a) Show that a polynomial \(P(x)\) that contains only odd powers of \(x\) is an odd function. (b) Show that a polynomial \(P(x)\) that contains only even powers of \(x\) is an even function. (c) Show that if a polynomial \(P(x)\) contains both odd and even powers of \(x\) , then it is neither an odd nor an even function. (d) Express the function $$P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5$$ as the sum of an odd function and an even function.

Step-by-Step Solution

Answer

P(x) can be expressed as the sum of odd part \(x^5 + 6x^3 - 2x\) and even part \(-x^2 + 5\).

Step 1: Understanding Odd Function

A function is odd if it satisfies the condition \(f(-x) = -f(x)\) for all real \(x\). This means that both sides of the function are symmetric about the origin.

Step 2: Proving Odd Polynomial

Consider a polynomial \(P(x) = a_n x^n + a_{n-2} x^{n-2} + \ldots + a_1 x\) where each exponent is odd. For this polynomial, \(P(-x) = a_n (-x)^n + a_{n-2} (-x)^{n-2} + \ldots + a_1 (-x)\), which simplifies to \(-a_n x^n - a_{n-2} x^{n-2} - \ldots - a_1 x = -P(x)\). Thus, the polynomial is odd.

Step 3: Understanding Even Function

A function is even if it satisfies the condition \(f(-x) = f(x)\) for all real \(x\). This indicates that both halves of the graph are symmetric about the y-axis.

Step 4: Proving Even Polynomial

Consider a polynomial \(P(x) = a_n x^n + a_{n-2} x^{n-2} + \ldots + a_0\) where each exponent is even. In this case, \(P(-x) = a_n (-x)^n + a_{n-2} (-x)^{n-2} + \ldots + a_0\), which simplifies to \(a_n x^n + a_{n-2} x^{n-2} + \ldots + a_0 = P(x)\). Hence, the polynomial is even.

Step 5: Combination of Odd and Even Powers

If a polynomial contains both odd and even powers, consider \(P(x) = f(x) + g(x)\) where \(f(x)\) is the part with odd powers and \(g(x)\) is the part with even powers. Then \(P(-x) = f(-x) + g(-x) = -f(x) + g(x)\) which does not equal \(f(x) + g(x)\) nor \(-[f(x) + g(x)]\). Thus, the function is neither odd nor even.

Step 6: Expressing a Polynomial as a Sum of Odd and Even Functions

To express \(P(x) = x^5 + 6x^3 - x^2 - 2x + 5\) as a sum of an odd function and an even function, separate it into odd and even terms: \(P(x) = (x^5 + 6x^3 - 2x) + (-x^2 + 5)\). The odd function is \(x^5 + 6x^3 - 2x\) and the even function is \(-x^2 + 5\).

Key Concepts

Odd Function

An odd function is one that shows a specific kind of symmetry. It's characterized by the property that for every value of \(x\), \(f(-x) = -f(x)\).
This means that the graph of the function is symmetric about the origin — whenever you reflect a point across the origin, it lands on another point of the graph.

• Think of it visually: if you rotate the graph 180 degrees around the origin, it will look the same.
• A real-world example is the sine function, \(\sin(x)\), which satisfies \(\sin(-x) = -\sin(x)\).

For polynomials, if only odd powers of \(x\) are present, such as \(x^3\) or \(x\), the function will be odd, like in \(x^5 + 6x^3 - 2x\). This satisfies the odd function condition because negating the input negates the whole expression.

Even Function

Even functions display a different type of symmetry compared to odd functions. An even function satisfies \(f(-x) = f(x)\) for all values of \(x\).
This means the graph of the function is symmetric along the y-axis.

• Imagine folding the graph along the y-axis; each point on one side folds directly onto a point on the other side.
• An example familiar to us is the cosine function, \(\cos(x)\), where \(\cos(-x) = \cos(x)\).

A polynomial that consists of even powers of \(x\), such as \(x^2\) or \(x^4\), will automatically satisfy this condition. So, a polynomial like \(-x^2 + 5\) is an even function.

Polynomial Functions

Polynomial functions are expressions composed of variables raised to various power levels, joined together by addition, subtraction, or both.
They can be expressed as a sum of terms, each consisting of a coefficient multiplied by a variable raised to an even or odd power.

• Examples include \(x^2 + 3x + 2\) or \(4x^3 - x\).
• Depending on whether the terms involve only odd or even powers of \(x\) (or both), they can exhibit different symmetries.

A polynomial with only odd-powered terms, like \(x^3\), is systematically odd.
Meanwhile, those with only even-powered terms, such as \(x^2\), are even. If a polynomial has both even and odd powers, like \(x^5 + x^2\), it isn't strictly odd or even, meaning it has no symmetrical properties like the ones seen with only even or odd terms.

Symmetric Functions

Symmetry in functions, particularly relating to even and odd functions, is all about how they relate graphs to axes and the origin.

• For symmetric functions about the y-axis (even functions), folding the graph along this vertical line shows equal halves.
• Odd functions, symmetrically about the origin, involve a reflection through the origin where every point \((x, f(x))\) corresponds to \((-x, -f(x))\).

This symmetry tells us about the fundamental nature of a function, especially crucial in graphing and understanding analytic properties.
When you combine terms of both symmetries, like in \(x^5 + 6x^3 - x^2 - 2x + 5\), you can decompose it into parts — odd parts grouped separately from even parts — showing that even complex expressions reveal simpler symmetric components, such as \((x^5 + 6x^3 - 2x)\) and \((-x^2 + 5)\).