Question

Let \(f(x)=a x^{2}+b x+c,\) where \(a, b\) and \(c\) are odd integers. If \(x\) is an integer, show that \(f(x)\) must be an odd integer.

Short answer

For any integer \(x\), \(f(x)\) must be an odd integer since combining both odd and even summation leads to consistent odd sums due to odd constants.

Step-by-step solution

Analyzing the function

Given the quadratic function is \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are odd integers.

Squaring an integer

Any integer \(x\) squared, \(x^2\), will result in an integer. If \(x = 2k\) (even), then \(x^2 = 4k^2\) (even). If \(x = 2k+1\) (odd), then \(x^2 = 4k^2 + 4k + 1\) (odd).

Multiplying by an odd integer

Since \(a\) is odd, multiplying \(a\) with \(x^2\) will produce an odd result if \(x^2\) is odd, and an even result if \(x^2\) is even. Thus, at least one of the terms resulting from this step remains consistent with the integer property of \(x^2\).

Evaluating \(bx\)

Given \(b\) and \(x\) are integers, \(bx\) will be odd when \(x\) is odd and will be even when \(x\) is even.

Combining the results

Summing \(ax^2\) (odd or even), \(bx\) (odd or even) and the odd constant term \(c\), the term \(ax^2 + bx + c\) necessarily sums up to an odd integer due to the structure of odd and even sums.

Verifying consistent results

No matter whether \(x\) is even or odd, \(ax^2 + bx + c\) consistently results in an odd integer. The sums involve terms where the odd constant \(c\) ensures that the end result of functioning sums between the types always results odd (only odd).

Key concepts

Odd Integer

An integer is odd if it cannot be evenly divided by 2. In other words, an odd integer always has a remainder of 1 when divided by 2. Examples of odd integers include \(-3, -1, 1, 3, 5,...\). Odd numbers have a unique property when squared or added.

• Squaring an odd integer will always result in another odd integer. For example, \((2k+1)^2 = 4k^2 + 4k + 1\), which is clearly odd.
• Adding two odd integers always results in an even integer. For example, \((-3) + 3 = 0\) or \(5 + 7 = 12\).

In our function \(f(x) = ax^2 + bx + c\), all coefficients \(a, b,\) and \(c\) are odd integers, which strongly influences the function's output based on the properties of odd numbers. Combining these properties helps to derive the final result more clearly.

Parity Arguments

Parity refers to whether an integer is even or odd. This concept is crucial in understanding quadratic functions like \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are odd integers.
Parity arguments help us predict the outcome when integers are manipulated within a function. Here are some important parity rules:

• Even \(x\): If \(x = 2k\) (even), then \(x^2 = 4k^2\) (even).
• Odd \(x\): If \(x = 2k+1\) (odd), then \(x^2 = 4k^2 + 4k + 1\) (odd).
• Multiplying an odd integer by an even integer will always produce an even product. Example, \(3 \times 4 = 12\).
• Multiplying two odd integers results in an odd product. Example, \(3 \times 5 = 15\).

Using these parity rules, we can better understand each term in \(ax^2 + bx + c\). This is why terms involving odd integer multiplication results in predictable parity outcomes.
By applying these principles, the function \(ax^2 + bx + c\) can be analyzed to show that, regardless of \(x\)'s parity, the result will always be odd.

Integer Properties

Integer properties include behaviors and characteristics of numbers that are whole and can be positive, negative, or zero. Quadratic functions with integer coefficients exhibit distinct behaviors that depend on these properties.
Here are some key integer properties relevant to \(f(x) = ax^2 + bx + c\):

• Closure: Integers are closed under addition, subtraction, and multiplication. This means combining any integers through these operations will always result in another integer.
• Distributive Property: For any integers \(a, b,\) and \(c\), the following holds - \(a(b + c) = ab + ac\).
• Associative Property: The way in which integers are grouped in addition or multiplication doesn’t change the outcome. Example, \((a + b) + c = a + (b + c)\).
• Additive Inverses: Every integer \((a\) has an additive inverse \((-a\) such that \(a + (-a) = 0\).

These properties are significant in the analysis of \(ax^2 + bx + c\). Each integer coefficient (\(a, b, c\)) and input (\(x\)) provides a clear outcome when these properties are applied.
Especially with quadratic functions, understanding how integers add, multiply, and combine ensures we're able to properly predict and verify the function's result as demonstrated.