Fill in the missing values to show that polynomials form a closed system under subtraction

Answer 1

To show that polynomials form a closed system under subtraction, we need to demonstrate that subtracting any two polynomials will result in another polynomial.

Let's consider two arbitrary polynomials:

P(x) = a₀xⁿ + a₁xⁿ⁻¹ + ... + aₙ₋₁x + aₙ
Q(x) = b₀xⁿ + b₁xⁿ⁻¹ + ... + bₙ₋₁x + bₙ

Now, let's subtract Q(x) from P(x):

P(x) - Q(x) = (a₀xⁿ + a₁xⁿ⁻¹ + ... + aₙ₋₁x + aₙ) - (b₀xⁿ + b₁xⁿ⁻¹ + ... + bₙ₋₁x + bₙ)

We can simplify this expression by subtracting each term individually:

= (a₀xⁿ - b₀xⁿ) + (a₁xⁿ⁻¹ - b₁xⁿ⁻¹) + ... + (aₙ₋₁x - bₙ₋₁x) + (aₙ - bₙ)

As we can see, each term in the resulting expression is still a polynomial term, either with the same degree or a lower degree. Therefore, the expression P(x) - Q(x) represents a polynomial.

In conclusion, subtracting any two polynomials results in another polynomial, which demonstrates that polynomials form a closed system under subtraction.