Problem 44 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Use a calculator to solve if necessary. An equation \(f(x)=0\) involves only odd powers of \(x\) with positive coefficients. Explain why this equation has no real root except \(x=0\).

Step-by-Step Solution

Answer

This equation has no real root except at \(x=0\) because \(f(x) > 0\) for \(x > 0\) and \(f(x) < 0\) for \(x < 0\).

Step 1: Understand Odd Powers of x

An equation involving only odd powers of \(x\) takes the form \(f(x) = a_1 x + a_3 x^3 + a_5 x^5 + \, ... \,\), where \(a_1, a_3, a_5, ...\) are positive coefficients. The crucial property of odd powers is that \((-x)^n = -x^n\) for any odd number \(n\).

Step 2: Analyze Behavior for x > 0

For \(x > 0\), each term \(a_n x^n\) (where \(n\) is odd) is positive. Therefore, \(f(x) > 0\) for all \(x > 0\) because it is a sum of positive numbers.

Step 3: Analyze Behavior for x < 0

For \(x < 0\), each term \(a_n x^n\) is negative because \((-x)^n = -x^n\) for odd \(n\). Consequently, \(f(x) < 0\) for all \(x < 0\), as it is a sum of negative numbers.

Step 4: Conclusion on Real Roots

Given that \(f(x) > 0\) for \(x > 0\) and \(f(x) < 0\) for \(x < 0\), the only potential real root where \(f(x) = 0\) can occur is at \(x = 0\). No other real value of \(x\) can satisfy \(f(x) = 0\).

Key Concepts

Polynomial Equation

A polynomial equation is formed by setting a polynomial equal to zero. The polynomial itself consists of terms made up of a variable, usually represented as \( x \), raised to whole number powers, each multiplied by coefficients.

• Example: \( f(x) = a_1 x + a_3 x^3 + a_5 x^5 + \, ... \,\)

This equation involves only odd powers of \( x \), which means each term has an odd exponent such as 1, 3, 5, etc.
These odd powers make understanding the behavior of the polynomial very important when determining solutions, or its roots.

Real Roots

Real roots are the values of \( x \) that satisfy the equation \( f(x) = 0 \). Finding real roots is essentially determining the x-values where the polynomial crosses the x-axis on a graph. To find real roots, we need to analyze the behavior of the polynomial both for positive and negative values of \( x \).
If an equation only has odd powers of \( x \) with positive coefficients, it handles positive and negative values of \( x \) distinctively:

• For \( x > 0 \), since each term is a positive number raised to an odd power times a positive coefficient, \( f(x) \) is always positive.
• For \( x < 0 \), every odd power results in a negative value; thus \( f(x) \) becomes a sum of negative terms.

Therefore, only at \( x = 0 \) the polynomial equation could potentially cross the x-axis, making \( x = 0 \) the sole real root.

Power of x

The power of \( x \) in a polynomial determines the term's behavior as \( x \) changes in values. In our specific equation, where only odd powers are present, we can observe:

• Odd exponents like 1, 3, 5, etc., play a crucial role in the behavior of the polynomial value.
• When \( x \) is negative, the odd power flips the sign of the number (for example, \((-x)^3 = -x^3\)).
• When \( x \) is positive, the output stays positive due to the multiplication with positive coefficients.

This flipping property of odd powers ensures that the sign of \( f(x) \) changes depending on whether \( x \) is negative or positive. This is why the equation cannot be zero for any \( x \) other than zero, because it can't switch from negative to positive or vice versa at any point other than crossing through zero.

Positive Coefficients

Positive coefficients play a significant role in the polynomial equation's overall behavior. They are the multipliers of the terms in the equation, ensuring that the output of the polynomial term aligns with expected behavior based on the sign of \( x \).

• For every term \( a_n x^n \) with \( n \) odd and \( a_n > 0 \), the result is positive when \( x > 0 \), contributing positively to \( f(x) \).
• Conversely, when \( x < 0 \), each term becomes negative, resulting in \( f(x) \) being negative.

This consistent behavior due to positive coefficients limits the possibilities of where \( f(x) \) can equal zero to just \( x=0 \).
Understanding the impact of positive coefficients alongside odd powers of \( x \) makes clear why the equation has only one real root.