Use the following multiplication problem to show how multiplying polynomials is a closed system.\left(x^2-1\right)\left(-2x+5\right)

Answer 1

To demonstrate that the multiplication of polynomials is a closed system using the given expression \((x^2 - 1)(-2x + 5)\), we will multiply the two polynomials together and show that the result is also a polynomial.

Step 1: Write Down the Polynomials

We have:

1. \( A = x^2 - 1 \)
2. \( B = -2x + 5 \)

Step 2: Use the Distributive Property

To multiply the two polynomials, we will apply the distributive property (also known as the FOIL method in the case of two binomials).

\[ (x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 - 1 \cdot (-2x) - 1 \cdot 5 \]

Step 3: Perform the Multiplication

Now, we will carry out each multiplication:

1. \( x^2 \cdot (-2x) = -2x^3 \)
2. \( x^2 \cdot 5 = 5x^2 \)
3. \( -1 \cdot (-2x) = 2x \)
4. \( -1 \cdot 5 = -5 \)

Step 4: Combine Like Terms

Now, we sum the results from the multiplications:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Final Expression

The final expression after combining the terms is:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Conclusion

The resulting expression \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial. Since we started with two polynomials and obtained another polynomial, this shows that the multiplication of polynomials is closed. In this case, if you multiply any two polynomials, the result will always be a polynomial as well. Thus, the set of all polynomials forms a closed system under multiplication.

Answer 2

can you do this in two sentences

Answer 3

Multiplying the polynomials \((x^2 - 1)\) and \((-2x + 5)\) yields \(-2x^3 + 5x^2 + 2x - 5\), which is also a polynomial. This demonstrates that the set of polynomials is closed under multiplication, as the product of two polynomials results in another polynomial.