Checking Algebraic Statements

We are asked to evaluate two algebraic statements and determine which one is correct. Let's analyze each statement individually.

Analysis of Statement I: \(x^3 + y^3 + z^3 = 360\)

Statement I gives us the values \(x = 12\), \(y = -2\), and \(z = -10\). We need to check if \(x^3 + y^3 + z^3\) equals 360.

First, let's calculate the sum \(x+y+z\):

\(x+y+z = 12 + (-2) + (-10)\)

\(x+y+z = 12 - 2 - 10\)

\(x+y+z = 10 - 10\)

\(x+y+z = 0\)

There is a useful algebraic identity that states: If \(x+y+z=0\), then \(x^3+y^3+z^3 = 3xyz\).

Since we found that \(x+y+z = 0\), we can use this identity to calculate \(x^3+y^3+z^3\):

\(x^3+y^3+z^3 = 3xyz\)

\(x^3+y^3+z^3 = 3 \times (12) \times (-2) \times (-10)\)

\(x^3+y^3+z^3 = 36 \times (20)\)

\(x^3+y^3+z^3 = 720\)

The statement claims that \(x^3 + y^3 + z^3 = 360\). Our calculation shows that \(x^3 + y^3 + z^3 = 720\). Since \(720 \neq 360\), Statement I is incorrect.

Analysis of Statement II: \(4x^2 + 4y^2 = 4480\)

Statement II gives us two conditions: \(x+y = 48\) and \(4xy = 128\). We need to check if \(4x^2 + 4y^2\) equals 4480.

From the second condition, \(4xy = 128\), we can find the value of \(xy\):

\(xy = \frac{128}{4}\)

\(xy = 32\)

We need to evaluate \(4x^2 + 4y^2\), which can be factored as \(4(x^2 + y^2)\).

We know the algebraic identity for the square of a sum: \((x+y)^2 = x^2 + 2xy + y^2\).

We can rearrange this identity to find \(x^2 + y^2\):

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

Now, substitute the given values \(x+y = 48\) and \(xy = 32\) into this expression:

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

Calculate \(48^2\):

\(48^2 = 48 \times 48 = 2304\)

Calculate \(2 \times 32\):

\(2 \times 32 = 64\)

Now, substitute these values back into the expression for \(x^2 + y^2\):

\(x^2 + y^2 = 2304 - 64\)

\(x^2 + y^2 = 2240\)

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

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

\(4(2240) = 8960\)

The statement claims that \(4x^2 + 4y^2 = 4480\). Our calculation shows that \(4x^2 + 4y^2 = 8960\). Since \(8960 \neq 4480\), Statement II is incorrect.

Conclusion on Algebraic Statements

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

Therefore, neither Statement I nor Statement II is correct.

Revision Table: Key Algebraic Identities

Understanding basic algebraic identities is crucial for solving problems like this. Here are some relevant identities:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a-b)(a+b)\)
• \((a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
• \(a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)\)
• If \(a+b+c = 0\), then \(a^3+b^3+c^3 = 3abc\)

Additional Information: Verifying Algebraic Statements

When asked to verify algebraic statements, it's important to follow a systematic approach:

1. Carefully read the statement and identify the given conditions or values.
2. Identify what the statement claims to be true.
3. Use the given information and relevant algebraic rules or identities to calculate the value of the expression in question.
4. Compare your calculated value with the value claimed in the statement.
5. Conclude whether the statement is correct or incorrect based on the comparison.

In Statement I, recognising the condition \(x+y+z=0\) is key to using the simpler identity \(x^3+y^3+z^3 = 3xyz\). Calculating \(x^3\), \(y^3\), and \(z^3\) separately and summing them up would also work but might be more calculation-intensive for larger numbers.

In Statement II, using the identity \((x+y)^2 = x^2 + 2xy + y^2\) to find \(x^2+y^2\) is a standard technique when \(x+y\) and \(xy\) are known. This avoids needing to find the individual values of \(x\) and \(y\) first (which would involve solving a quadratic equation).