Let's break down the process of subtracting two polynomials step-by-step to show that polynomials are closed under subtraction.
Given the two polynomials:
[tex]\[ P(x) = 5x^2 + 3x + 4 \][/tex]
[tex]\[ Q(x) = 2x^2 + 5x - 1 \][/tex]
we want to perform the subtraction [tex]\( P(x) - Q(x) \)[/tex].
1. Identify the corresponding coefficients for each term in both polynomials:
- Coefficients of [tex]\( x^2 \)[/tex]: [tex]\( 5 \)[/tex] (from [tex]\( P(x) \)[/tex]) and [tex]\( 2 \)[/tex] (from [tex]\( Q(x) \)[/tex]).
- Coefficients of [tex]\( x \)[/tex]: [tex]\( 3 \)[/tex] (from [tex]\( P(x) \)[/tex]) and [tex]\( 5 \)[/tex] (from [tex]\( Q(x) \)[/tex]).
- Constant terms: [tex]\( 4 \)[/tex] (from [tex]\( P(x) \)[/tex]) and [tex]\( -1 \)[/tex] (from [tex]\( Q(x) \)[/tex]).
2. Subtract the coefficients term-by-term:
- Coefficient of [tex]\( x^2 \)[/tex] term: [tex]\( 5 - 2 = 3 \)[/tex].
- Coefficient of [tex]\( x \)[/tex] term: [tex]\( 3 - 5 = -2 \)[/tex].
- Constant term: [tex]\( 4 - (-1) = 4 + 1 = 5 \)[/tex].
So, the resulting polynomial from the subtraction is:
[tex]\[ 3x^2 - 2x + 5 \][/tex]
To address the question of whether the result is a polynomial:
- The result, [tex]\( 3x^2 - 2x + 5 \)[/tex], is indeed a polynomial. Polynomials are mathematical expressions involving a sum of powers of [tex]\( x \)[/tex] with coefficients. The subtraction of two polynomials results in another polynomial because the operations do not change the form or the nature of the polynomial.
Given the choices, the correct one is:
[tex]\[ 3x^2 - 2x + 5 ; \text{ will be a polynomial} \][/tex]
Thus, the correct answer is:
[tex]\[ 3 x^2 - 2 x + 5 ; \text{ will be a polynomial} \][/tex]
