Question

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)
(
𝑥
−
5
)
(
𝑥
+
3
)


Expression 2: 2x2−x2−2x−20+5

Answer 1

To prove that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we'll first simplify Expression 1 using the Distributive Property (FOIL method) and then simplify Expression 2.

### Expression 1: \((x - 5)(x + 3)\)

Using the FOIL method (First, Outside, Inside, Last):

1. **First:** \(x \cdot x = x^2\)
2. **Outside:** \(x \cdot 3 = 3x\)
3. **Inside:** \(-5 \cdot x = -5x\)
4. **Last:** \(-5 \cdot 3 = -15\)

Now, combine the results:
\[
x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]

### Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)

Now we simplify Expression 2 step-by-step:
1. Combine like terms:
- \(2x^2 - x^2 = x^2\)
- \(-2x\) remains the same.
- \(-20 + 5 = -15\)

So, we have:
\[
x^2 - 2x - 15
\]

### Conclusion

Both expressions simplify to \(x^2 - 2x - 15\), which proves they are equivalent.

Thus, we can conclude:
- \((x - 5)(x + 3)\) simplifies to \(x^2 - 2x - 15\).
- \(2x^2 - x^2 - 2x - 20 + 5\) also simplifies to \(x^2 - 2x - 15\).

Since both expressions simplify to the same polynomial, they are equivalent.