Understanding Proportionality in Algebra

The question deals with the concept of proportionality between algebraic expressions. When we say an expression A is proportional to an expression B, it means that A is equal to a constant multiple of B. Mathematically, this is written as \(A \propto B\), which is equivalent to \(A = k \cdot B\), where \(k\) is a constant of proportionality.

We are given that \((a^3 + b^3)\) is proportional to \((a^2 – b^2)\). Let's write this relationship using a constant \(k\):

\[(a^3 + b^3) = k(a^2 – b^2)\]

Our goal is to find out what \((a^2 - ab + b^2)\) is proportional to. To do this, we can use the factorization formulas for the sum of cubes and the difference of squares.

Factorizing the Expressions

Recall the important algebraic factorization formulas:

• Sum of Cubes: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
• Difference of Squares: \(a^2 - b^2 = (a-b)(a+b)\)

Now, let's substitute these factorized forms into our proportionality equation:

\[(a+b)(a^2 - ab + b^2) = k(a-b)(a+b)\]

Solving for the Relationship

We have the equation \((a+b)(a^2 - ab + b^2) = k(a-b)(a+b)\). Assuming that \((a+b)\) is not equal to zero (if \(a+b=0\), the original expression \(a^2-b^2\) would be 0, leading to potential issues or trivial cases), we can divide both sides of the equation by \((a+b)\).

Dividing both sides by \((a+b)\), we get:

\[\frac{(a+b)(a^2 - ab + b^2)}{(a+b)} = \frac{k(a-b)(a+b)}{(a+b)}\]

This simplifies to:

\[(a^2 - ab + b^2) = k(a-b)\]

Identifying the Proportionality

The resulting equation \((a^2 - ab + b^2) = k(a-b)\) shows that the expression \((a^2 - ab + b^2)\) is equal to a constant \(k\) multiplied by the expression \((a-b)\). By the definition of proportionality, this means that \((a^2 - ab + b^2)\) is proportional to \((a-b)\).

Let's look at the given options:

1. (a - b)
2. (a + b)
3. (a + ab + b)
4. (a³ – b³)

Comparing our result \((a^2 - ab + b^2) = k(a-b)\) with the options, we see that \((a^2 - ab + b^2)\) is proportional to \((a - b)\).

Thus, the correct option is (a - b).

Revision Table: Key Algebra Concepts

Additional Information: Conditions and Special Cases

In our solution, we divided by \((a+b)\). This step is valid only if \((a+b) \neq 0\), which means \(a \neq -b\). If \(a = -b\), then \(a+b = 0\). In this case:

• \(a^3 + b^3 = a^3 + (-a)^3 = a^3 - a^3 = 0\)
• \(a^2 - b^2 = a^2 - (-a)^2 = a^2 - a^2 = 0\)

The initial condition \((a^3 + b^3) \propto (a^2 – b^2)\) becomes \(0 \propto 0\), which is true for any constant \(k\). The expression \((a^2 - ab + b^2)\) becomes \(a^2 - a(-a) + (-a)^2 = a^2 + a^2 + a^2 = 3a^2\).

And \((a-b)\) becomes \(a - (-a) = 2a\).

So, for \(a = -b\), we have \(3a^2\) proportional to \(2a\). This means \(3a^2 = k(2a)\). If \(a \neq 0\), then \(3a = 2k\), so \(k = \frac{3a}{2}\). The constant of proportionality \(k\) depends on \(a\) (or \(b\)), which contradicts the idea of \(k\) being a fixed constant for the general proportionality relationship.

Therefore, the proportionality \((a^2 - ab + b^2) = k(a-b)\) holds generally when \((a+b) \neq 0\). The question implicitly assumes the proportionality holds for generic values of \(a\) and \(b\) where the expressions are non-zero.