Which of the following equations is an example showing that polynomials form a closed system under multiplication?(1 point)

Question

Responses

(x2+1)(x−12)=x3−12x2+x−12
(
𝑥
2
+
1
)
(
𝑥
−
1
2
)
=
𝑥
3
−
1
2
𝑥
2
+
𝑥
−
1
2
open paren x squared plus 1 close paren times open paren x minus 1 half close paren is equal to x cubed minus 1 half x squared plus x minus 1 half

Answer 1

To determine if the equation shows that polynomials form a closed system under multiplication, we need to ensure that the product of two polynomials results in another polynomial.

In your given equation:

\[ (x^2 + 1)(x - \frac{1}{2}) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \]

Let's analyze the left side:

1. \(x^2 + 1\) is a polynomial of degree 2.
2. \(x - \frac{1}{2}\) is a polynomial of degree 1.

When we multiply these two polynomials together, the degree of the resulting polynomial will be the sum of the degrees of the two multiplicands, which is \(2 + 1 = 3\).

Now, let’s look at the right side:

• The expression \(x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}\) is indeed a polynomial of degree 3.

Since both sides of the equation are polynomials, this illustrates that the product of two polynomials (in this case, \(x^2 + 1\) and \(x - \frac{1}{2}\)) results in another polynomial. Therefore, this equation serves as an example showing that polynomials form a closed system under multiplication.

Conclusion: The equation \( (x^2 + 1)(x - \frac{1}{2}) = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \) is indeed an example of polynomials' closure under multiplication.