Understanding the Algebraic Identity for \(a^3 - b^3\)

This question asks for the value of the expression \(a^3 - b^3\) given that the difference between 'a' and 'b' (\(a - b\)) is equal to 1. To solve this, we need to use a fundamental algebraic identity related to the difference of cubes.

The key algebraic identity for the difference of two cubes, \(a^3 - b^3\), is:

\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

This identity shows that the difference of cubes can be factored into two parts: the difference of the terms (\(a - b\)) and a quadratic expression (\(a^2 + ab + b^2\)).

Applying the Given Information to Find \(a^3 - b^3\)

We are given that \(a - b = 1\). We can substitute this value directly into the algebraic identity for \(a^3 - b^3\):

Substitute \(a - b = 1\) into the identity:

\(a^3 - b^3 = (\mathbf{a - b})(a^2 + ab + b^2)\)

\(a^3 - b^3 = (\mathbf{1})(a^2 + ab + b^2)\)

Multiplying any expression by 1 results in the same expression. Therefore, simplifying the equation gives us:

\(a^3 - b^3 = a^2 + ab + b^2\)

Comparing with the Options

Now, let's compare our derived value for \(a^3 - b^3\) with the given options:

• Option 1: \(a^2 + ab + b^2\)
• Option 2: \((a + b)^2 + 3ab\)
• Option 3: \(a^2 - ab + b^2\)
• Option 4: \(a^2 + 2ab + b^2\)

Our calculated value, \(a^3 - b^3 = a^2 + ab + b^2\), exactly matches Option 1.

Conclusion

Given that \(a - b = 1\), the value of \(a^3 - b^3\) is found by using the algebraic identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) and substituting \(a - b = 1\). This substitution leads to \(a^3 - b^3 = 1 \times (a^2 + ab + b^2) = a^2 + ab + b^2\).

The correct value of \(a^3 - b^3\) when \(a - b = 1\) is \(a^2 + ab + b^2\).

Revision Table: Key Algebraic Identities

Additional Information: Related Algebraic Expansions

While the difference of cubes identity was direct, sometimes you might need to relate \(a^3 - b^3\) to \((a - b)^3\) or \((a + b)^3\). Let's look at the expansion of \((a - b)^3\):

\((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)

Rearranging the terms to isolate \(a^3 - b^3\):

\(a^3 - b^3 = (a - b)^3 + 3a^2b - 3ab^2\)

\(a^3 - b^3 = (a - b)^3 + 3ab(a - b)\)

Using the given \(a - b = 1\):

\(a^3 - b^3 = (1)^3 + 3ab(1)\)

\(a^3 - b^3 = 1 + 3ab\)

This gives us an alternative expression for \(a^3 - b^3\) when \(a - b = 1\). Does this relate to our answer \(a^2 + ab + b^2\)? Yes, it does, but relating \(1 + 3ab\) back to \(a^2 + ab + b^2\) requires more steps or additional information about 'a' and 'b'. The identity method is the most direct route here.

The identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) is crucial for problems involving difference of cubes and given differences or sums of the base terms.