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Which of the following shows that polynomials are closed under subtraction when polynomial [tex]$5x-6$[/tex] is subtracted from [tex]$3x^2-6x+2$[/tex]?

A. [tex][tex]$3x^2-11x+8$[/tex][/tex] may or may not be a polynomial.
B. [tex]$3x^2-11x+8$[/tex] will be a polynomial.
C. [tex]$3x^2-x+4$[/tex] may or may not be a polynomial.
D. [tex][tex]$3x^2-x+4$[/tex][/tex] will be a polynomial.

Sagot :

GinnyAnsweridn
To determine if polynomials are closed under subtraction using the given polynomial expressions, let's perform the subtraction step by step.

### Given Polynomials:
1. [tex]\( 3x^2 - 6x + 2 \)[/tex]
2. [tex]\( 5x - 6 \)[/tex]

### Goal:
Subtract the second polynomial from the first polynomial, i.e., [tex]\( (3x^2 - 6x + 2) - (5x - 6) \)[/tex].

### Step-by-Step Subtraction:

1. Rewrite the subtraction as adding the opposite:
[tex]\[(3x^2 - 6x + 2) - (5x - 6) = 3x^2 - 6x + 2 - 5x + 6\][/tex]

2. Combine like terms:
[tex]\[3x^2 - 6x - 5x + 2 + 6\][/tex]

3. Simplify:
[tex]\[3x^2 - 11x + 8\][/tex]

### Result:
The result of the subtraction is [tex]\(3x^2 - 11x + 8\)[/tex].

### Checking if the Result is a Polynomial:

- A polynomial is an expression that consists of variables and coefficients, involving only non-negative integer exponents of variables.
- The resulting expression [tex]\(3x^2 - 11x + 8\)[/tex] meets these criteria:
- The term [tex]\( 3x^2 \)[/tex] is a polynomial term (degree 2).
- The term [tex]\( -11x \)[/tex] is a polynomial term (degree 1).
- The term [tex]\( 8 \)[/tex] is a polynomial term (degree 0).

Thus, [tex]\(3x^2 - 11x + 8\)[/tex] is indeed a polynomial.

### Conclusion:
Based on the given options, the correct answer is:

b. [tex]\(3x^2 - 11x + 8\)[/tex] will be a polynomial.

This confirms that polynomials are closed under subtraction.
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