Answer :
Let's simplify the given expression and demonstrate the closure property:
The expression is: [tex]\((2x^3 + x^2 - 4x) - (9x^3) - (-7x^3 + 4x^2 - 4x)\)[/tex].
To simplify, we need to:
1. Distribute the negative sign to the terms inside the parentheses.
2. Combine like terms.
Start with the expression:
1. Distribute the negative signs:
[tex]\[
(2x^3 + x^2 - 4x) - 9x^3 + 7x^3 - 4x^2 + 4x
\][/tex]
2. Combine like terms:
- For [tex]\(x^3\)[/tex]: [tex]\(2x^3 - 9x^3 + 7x^3 = 0x^3 = 0\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(x^2 - 4x^2 = -3x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-4x + 4x = 0x = 0\)[/tex]
The simplified expression is:
[tex]\[
-3x^2
\][/tex]
Conclusion: The result [tex]\(-3x^2\)[/tex] is a polynomial, showing that the closure property of polynomials holds because the subtraction of two polynomials results in another polynomial. Thus, the correct simplification is [tex]\(-3x^2\)[/tex], and it is indeed a polynomial.
The expression is: [tex]\((2x^3 + x^2 - 4x) - (9x^3) - (-7x^3 + 4x^2 - 4x)\)[/tex].
To simplify, we need to:
1. Distribute the negative sign to the terms inside the parentheses.
2. Combine like terms.
Start with the expression:
1. Distribute the negative signs:
[tex]\[
(2x^3 + x^2 - 4x) - 9x^3 + 7x^3 - 4x^2 + 4x
\][/tex]
2. Combine like terms:
- For [tex]\(x^3\)[/tex]: [tex]\(2x^3 - 9x^3 + 7x^3 = 0x^3 = 0\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(x^2 - 4x^2 = -3x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-4x + 4x = 0x = 0\)[/tex]
The simplified expression is:
[tex]\[
-3x^2
\][/tex]
Conclusion: The result [tex]\(-3x^2\)[/tex] is a polynomial, showing that the closure property of polynomials holds because the subtraction of two polynomials results in another polynomial. Thus, the correct simplification is [tex]\(-3x^2\)[/tex], and it is indeed a polynomial.
