Question

Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in one sentence for each expression and show your work.
Expression 1: (x−5)(x+3)
Expression 2: 2x^2−x^2-2x-20+5

Answer 1

In 1–2 sentences, explain why the following operation is a closed system under subtraction.
(9x^2+17x+13)−(2x^2+12x+11)

Answer 2

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

Answer 3

To prove that the given algebraic expressions are equivalent using the Distributive Property, we will simplify both expressions and show that they are identical.

**Expression 1:**
\[
(x - 5)(x + 3)
\]
Using the Distributive Property (also known as the FOIL method for binomials), we calculate:
\[
= x(x + 3) - 5(x + 3)
\]
\[
= x^2 + 3x - 5x - 15
\]
Combining like terms:
\[
= x^2 - 2x - 15
\]

Now, let's simplify **Expression 2**:
\[
2x^2 - x^2 - 2x - 20 + 5
\]
First, simplify \(2x^2 - x^2\):
\[
= x^2 - 2x - 20 + 5
\]
Now, combine the constants \(-20 + 5\):
\[
= x^2 - 2x - 15
\]

Now we compare both simplified expressions:
- **Expression 1:** \(x^2 - 2x - 15\)
- **Expression 2:** \(x^2 - 2x - 15\)

Since both expressions are equal, we conclude that they are equivalent.

**Conclusion:**
The two expressions are equivalent because when simplified, both yield the same polynomial expression \(x^2 - 2x - 15\).

Answer 4

The operation is a closed system under subtraction because subtracting one polynomial from another results in another polynomial, maintaining the same structure of polynomial expressions. In this case, subtracting \((2x^2 + 12x + 11)\) from \((9x^2 + 17x + 13)\) yields a polynomial of degree 2, confirming that the set of polynomials is closed under the operation of subtraction.

Answer 5

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

**Step 1: Apply the Distributive Property**
We will distribute each term in the first polynomial to each term in the second polynomial:

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

**Step 2: Simplify each term**
Calculating each term individually:
- \(x^2 \cdot -2x = -2x^3\)
- \(x^2 \cdot 5 = 5x^2\)
- \(-1 \cdot -2x = 2x\)
- \(-1 \cdot 5 = -5\)

Now, we collect all these results together:

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

**Step 3: Combine the terms**
Since there are no like terms to combine, we leave the expression as is:

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

**Conclusion:**
The resulting expression \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial of degree 3. Thus, we have shown that multiplying the polynomials \((x^2 - 1)\) and \((-2x + 5)\) results in another polynomial, confirming that the set of polynomials is closed under multiplication.