Calculating the Value of \( a^3 + b^3 + c^3 - 3abc \)

The question asks for the value of the expression \( a^3 + b^3 + c^3 - 3abc \) given the values of \( a + b + c \) and \( ab + bc + ca \). This type of problem frequently uses algebraic identities, which are fundamental in algebra.

Understanding the Polynomial Identity for \( a^3 + b^3 + c^3 - 3abc \)

There is a standard algebraic identity that relates \( a^3 + b^3 + c^3 - 3abc \) to the sum and product of the variables. The identity is:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \]

To find the value of the expression \( a^3 + b^3 + c^3 - 3abc \), we need to know the values of \( (a + b + c) \) and \( (a^2 + b^2 + c^2 - ab - bc - ca) \).

Finding the Value of \( a^2 + b^2 + c^2 \)

We are given \( a + b + c = 5 \) and \( ab + bc + ca = 10 \).

We can find the value of \( a^2 + b^2 + c^2 \) using another common algebraic identity:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

Rearranging this formula to solve for \( a^2 + b^2 + c^2 \):

\[ a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) \]

Now, we can substitute the given values:

• \( a + b + c = 5 \)
• \( ab + bc + ca = 10 \)

So, the value of \( a^2 + b^2 + c^2 \) is:

\[ a^2 + b^2 + c^2 = (5)^2 - 2(10) \]

\[ a^2 + b^2 + c^2 = 25 - 20 \]

\[ a^2 + b^2 + c^2 = 5 \]

Calculating the Second Factor: \( a^2 + b^2 + c^2 - ab - bc - ca \)

Now that we have the value of \( a^2 + b^2 + c^2 \), we can calculate the second factor needed for the \( a^3 + b^3 + c^3 - 3abc \) formula. The second factor is \( a^2 + b^2 + c^2 - ab - bc - ca \).

We can write this as \( (a^2 + b^2 + c^2) - (ab + bc + ca) \). We know:

• \( a^2 + b^2 + c^2 = 5 \)
• \( ab + bc + ca = 10 \)

Substituting these values:

\[ a^2 + b^2 + c^2 - ab - bc - ca = 5 - 10 \]

\[ a^2 + b^2 + c^2 - ab - bc - ca = -5 \]

Finding the Final Value of \( a^3 + b^3 + c^3 - 3abc \)

Finally, we can use the main identity for \( a^3 + b^3 + c^3 - 3abc \):

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \]

We have the values for both factors:

• \( a + b + c = 5 \)
• \( a^2 + b^2 + c^2 - ab - bc - ca = -5 \)

Substitute these values into the identity:

\[ a^3 + b^3 + c^3 - 3abc = (5)(-5) \]

\[ a^3 + b^3 + c^3 - 3abc = -25 \]

Thus, the value of \( a^3 + b^3 + c^3 - 3abc \) is -25. This demonstrates a typical application of algebraic identities in solving problems involving sums and products of variables.

This question tests understanding of fundamental algebraic identities and their application in finding the value of expressions like \( a^3 + b^3 + c^3 - 3abc \).