Question

select the correct answer.

let \\(f(x)\\) and \\(g(x)\\) be polynomials as shown below.

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$$\begin{aligned} f(x) &= a_0 + a_1x + a_2x^2 \\dots + a_nx^n \\\\ g(x) &= b_0 + b_1x + b_2x^2 \\dots + b_mx^m \\end{aligned}$$

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which of the following is true about \\(f(x)\\) and \\(g(x)\\)?

• \\(f(x)\\) and \\(g(x)\\) are closed under subtraction because when subtracted, the result will be a polynomial.
• \\(f(x)\\) and \\(g(x)\\) are closed under subtraction because when subtracted, the result will not be a polynomial.
• \\(f(x)\\) and \\(g(x)\\) are not closed under subtraction because when subtracted, the result will be a polynomial.
• \\(f(x)\\) and \\(g(x)\\) are not closed under subtraction because when subtracted, the result will not be a polynomial.

Explanation:

🆕 New Concept Discovered: Closure Property of Polynomials
Operations on polynomials always yield polynomials

Step 1: Understand the concept of closure

A set of numbers or expressions is "closed" under an operation (like addition, subtraction, or multiplication) if performing that operation on any members of the set always results in another member of that same set.

For polynomials, closure under subtraction means that if you subtract one polynomial from another, the result must always be a polynomial.

Step 2: Analyze the subtraction of two polynomials

Let \( f(x) \) and \( g(x) \) be the polynomials:

$$ f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n $$

$$ g(x) = b_0 + b_1x + b_2x^2 + \dots + b_mx^m $$

When we subtract \( g(x) \) from \( f(x) \), we combine like terms by subtracting their coefficients:

$$ f(x) - g(x) = (a_0 - b_0) + (a_1 - b_1)x + (a_2 - b_2)x^2 + \dots $$

Since the coefficients are real numbers, subtracting them results in new real number coefficients. The powers of \( x \) remain non-negative integers. Therefore, the resulting expression is guaranteed to be another polynomial.

Step 3: Match with the given options

Because subtracting any two polynomials always results in a polynomial, polynomials are closed under subtraction.

This matches the first option:

• \( f(x) \) and \( g(x) \) are closed under subtraction because when subtracted, the result will be a polynomial.

Answer:

\( f(x) \) and \( g(x) \) are closed under subtraction because when subtracted, the result will be a polynomial.