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Which statement shows how the two polynomials [tex]3x + 6[/tex] and [tex]5x^2 - 4x[/tex] demonstrate the closure property when multiplied?

A. [tex]15x^3 + 18x^2 - 24x[/tex] may or may not be a polynomial.
B. [tex]15x^3 + 18x^2 - 24x[/tex] is a polynomial.
C. [tex]15x^3 + 42x^2 - 24x[/tex] may or may not be a polynomial.
D. [tex]15x^3 + 42x^2 - 24x[/tex] is a polynomial.


Sagot :

To determine which statement correctly demonstrates the closure property under multiplication for the polynomials [tex]\(3x + 6\)[/tex] and [tex]\(5x^2 - 4x\)[/tex], let's verify the result of multiplying these polynomials.

When you multiply two polynomials, the result should also be a polynomial. Here is the step-by-step process of multiplying [tex]\(3x + 6\)[/tex] and [tex]\(5x^2 - 4x\)[/tex]:

1. Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[ (3x + 6)(5x^2 - 4x) \][/tex]

2. Multiply the terms:
[tex]\[ 3x \cdot 5x^2 = 15x^3 \][/tex]
[tex]\[ 3x \cdot (-4x) = -12x^2 \][/tex]
[tex]\[ 6 \cdot 5x^2 = 30x^2 \][/tex]
[tex]\[ 6 \cdot (-4x) = -24x \][/tex]

3. Combine the resulting terms:
[tex]\[ 15x^3 - 12x^2 + 30x^2 - 24x \][/tex]

4. Combine like terms:
[tex]\[ 15x^3 + (30x^2 - 12x^2) - 24x = 15x^3 + 18x^2 - 24x \][/tex]

Therefore, the product of [tex]\(3x + 6\)[/tex] and [tex]\(5x^2 - 4x\)[/tex] is [tex]\(15x^3 + 18x^2 - 24x\)[/tex], which is a polynomial. According to the closure property of polynomials, the product of two polynomials is also a polynomial.

The correct statement is:
[tex]\[ 15x^3 + 18x^2 - 24x \text{ is a polynomial} \][/tex]