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Use the following multiplication problem to show how multiplying polynomials is a closed system.

(x^2−1)(−2x+5)

3 answers

To demonstrate that multiplying polynomials is a closed system, we'll compute the product \((x^2 - 1)(-2x + 5)\) and show that the result is also a polynomial.

Step 1: Distribute each term in the first polynomial by each term in the second polynomial.

We can use the distributive property (also known as the FOIL method in this binomial case) to compute:

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

We will distribute each term in the first polynomial to each term in the second polynomial:

  1. Multiply \(x^2\) by \(-2x\): \[ x^2 \cdot (-2x) = -2x^3 \]

  2. Multiply \(x^2\) by \(5\): \[ x^2 \cdot 5 = 5x^2 \]

  3. Multiply \(-1\) by \(-2x\): \[ -1 \cdot (-2x) = 2x \]

  4. Multiply \(-1\) by \(5\): \[ -1 \cdot 5 = -5 \]

Step 2: Combine all the results.

Now, we combine all the products we calculated:

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

Step 3: Write the final result.

Thus, the product of the polynomials \( (x^2 - 1)(-2x + 5) \) is:

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

Conclusion:

The result \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial. Since we started with two polynomials and ended with another polynomial, we can conclude that the multiplication of polynomials is closed under the operation. This confirms that multiplying polynomials is indeed a closed system.

can you make the steps shorter please

Certainly! Here’s a more concise version of multiplying the polynomials \((x^2 - 1)(-2x + 5)\):

Step 1: Use the distributive property. Distribute each term:

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

Step 2: Calculate each term:

  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 3: Combine the results: \[ -2x^3 + 5x^2 + 2x - 5 \]

Conclusion: The final result \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial, demonstrating that the multiplication of polynomials is a closed system.