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.

Question

Expression 1: (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 using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

Step 1: Expand Expression 1

$$(x-5)(x+3)=x(x+3)-5(x+3)$$

Distributing $x$ and $-5$:

$$=x^2+3x-5x-15$$

Combine like terms:

$$=x^2-2x-15$$

Step 2: Simplify Expression 2

Expression 2 is given as:

$$2x^2-x^2-2x-20+5$$

Combine like terms:

$$=(2x^2-x^2)-2x-20+5$$

This simplifies to:

$$=x^2-2x-15$$

Conclusion:

Now we can see that both expressions equal $x^2-2x-15$. Thus, we have shown that the expressions $(x-5)(x+3)$ and $2x^2-x^2-2x-20+5$ are equivalent because they simplify to the same polynomial $x^2-2x-15$.

Answer 2

turn it all into words