Analyzing Algebraic Statements

Let's carefully analyze each statement provided in the question to determine its correctness.

Analysis of Statement I: Evaluating $x^3 + y^3 + z^3$

Statement I gives the values $x = 12$, $y = -2$, and $z = -10$. It claims that $x^3 + y^3 + z^3 = 720$ for these values.

We can evaluate this expression directly or use a relevant algebraic identity. A useful identity for the sum of cubes involves the sum of the variables: $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$.

A special case of this identity is when $x+y+z=0$. In this case, the right side becomes $0$, leading to $x^3 + y^3 + z^3 = 3xyz$.

Let's check the sum of the given values of $x$, $y$, and $z$:

$$x+y+z = 12 + (-2) + (-10) = 12 - 2 - 10 = 12 - 12 = 0$$

Since $x+y+z = 0$, we can use the identity $x^3 + y^3 + z^3 = 3xyz$.

Now, let's calculate $3xyz$ using the given values:

$$3xyz = 3 \times (12) \times (-2) \times (-10)$$

$$3xyz = 3 \times 12 \times ((-2) \times (-10))$$

$$3xyz = 3 \times 12 \times 20$$

$$3xyz = 36 \times 20$$

$$3xyz = 720$$

So, for $x=12$, $y=-2$, and $z=-10$, we found that $x^3 + y^3 + z^3 = 720$. This matches the claim made in Statement I.

Therefore, Statement I is correct.

Analysis of Statement II: Evaluating $4x^2 + 4y^2$

Statement II provides two pieces of information: $x+y = 48$ and $4xy = 128$. It asks for the value of $4x^2 + 4y^2$.

First, let's simplify the second equation to find the value of $xy$:

$$4xy = 128$$

Divide both sides by 4:

$$xy = \frac{128}{4}$$

$$xy = 32$$

We need to find the value of $4x^2 + 4y^2$. We can factor out 4:

$$4x^2 + 4y^2 = 4(x^2 + y^2)$$

Now, we need to find the value of $x^2 + y^2$. We know the algebraic identity $(x+y)^2 = x^2 + y^2 + 2xy$. We can rearrange this identity to solve for $x^2 + y^2$:

$$x^2 + y^2 = (x+y)^2 - 2xy$$

We are given that $x+y = 48$ and we found that $xy = 32$. Substitute these values into the equation for $x^2 + y^2$:

$$x^2 + y^2 = (48)^2 - 2(32)$$

Calculate $(48)^2$:

$$48^2 = (50 - 2)^2 = 50^2 - 2 \times 50 \times 2 + 2^2 = 2500 - 200 + 4 = 2304$$

Calculate $2(32)$:

$$2(32) = 64$$

Now, substitute these values back into the equation for $x^2 + y^2$:

$$x^2 + y^2 = 2304 - 64$$

$$x^2 + y^2 = 2240$$

Finally, calculate $4(x^2 + y^2)$:

$$4(x^2 + y^2) = 4 \times 2240$$

$$4 \times 2240 = 8960$$

So, the value of $4x^2 + 4y^2$ is 8960. This matches the claim made in Statement II.

Therefore, Statement II is correct.

Conclusion

Based on our analysis, both Statement I and Statement II are correct.

Statement I was verified using the special case of the sum of cubes identity when the sum of the variables is zero ($x+y+z=0$). Statement II was verified by using the identity for $(x+y)^2$ to find $x^2+y^2$ and then substituting the given values.

Revision Table: Key Algebraic Concepts

Additional Information: Using Algebraic Identities

Algebraic identities are powerful tools in simplifying expressions and solving equations. They are equations that are true for all values of the variables involved.

• Using identities can often save time compared to direct calculation, especially with larger numbers or complex expressions.
• Recognizing the form of an expression allows you to apply the appropriate identity. For instance, seeing a sum of three cubes prompts you to consider the $x^3+y^3+z^3$ identity.
• For expressions involving $x^2+y^2$ when $x+y$ and $xy$ are known, the identity $(x+y)^2 = x^2+y^2+2xy$ is fundamental.
• Practicing with different algebraic problems helps in mastering the application of these identities.

Understanding these core identities is crucial for success in algebra and related fields of mathematics.