To demonstrate how two polynomials [tex]\(4x + 6\)[/tex] and [tex]\(2x^2 - 8x\)[/tex] exhibit the closure property when multiplied, we need to perform the multiplication of the polynomials step-by-step and ensure that the result is also a polynomial.
Let's start by multiplying the polynomials:
[tex]\[ (4x + 6) \times (2x^2 - 8x) \][/tex]
First, distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[ 4x \times (2x^2 - 8x) + 6 \times (2x^2 - 8x) \][/tex]
Now calculate each product individually:
1. [tex]\( 4x \times 2x^2 = 8x^3 \)[/tex]
2. [tex]\( 4x \times -8x = -32x^2 \)[/tex]
3. [tex]\( 6 \times 2x^2 = 12x^2 \)[/tex]
4. [tex]\( 6 \times -8x = -48x \)[/tex]
Next, we combine all these terms together:
[tex]\[ 8x^3 - 32x^2 + 12x^2 - 48x \][/tex]
Now, combine the like terms:
[tex]\[ 8x^3 + (-32x^2 + 12x^2) - 48x \][/tex]
[tex]\[ 8x^3 - 20x^2 - 48x \][/tex]
The resulting polynomial from multiplying [tex]\(4x + 6\)[/tex] and [tex]\(2x^2 - 8x\)[/tex] is:
[tex]\[ 8x^3 - 20x^2 - 48x \][/tex]
This result is indeed a polynomial, specifically of degree 3. Therefore, the correct statement that shows how these two polynomials demonstrate the closure property when multiplied is:
[tex]\(8 x^3-20 x^2-48 x\)[/tex] is a polynomial.
