Evaluating Algebraic Expressions Using Identities

The problem asks us to find the value of the expression \(x^3 + y^3 - 9xy + 9\), given the condition that \(x + y + 3 = 0\). This problem can be solved by recognizing and applying a useful algebraic identity related to the sum of cubes.

Understanding the Given Condition

The condition is \(x + y + 3 = 0\). This can be rearranged to \(x + y = -3\).

Applying the Relevant Algebraic Identity

We know a powerful algebraic identity: For any numbers \(a\), \(b\), and \(c\),

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

A special case of this identity occurs when \(a + b + c = 0\). In this case, the right side of the identity becomes \((0)(a^2 + b^2 + c^2 - ab - bc - ca) = 0\).

So, if \(a + b + c = 0\), then \(a^3 + b^3 + c^3 - 3abc = 0\), which means \(a^3 + b^3 + c^3 = 3abc\).

Connecting the Problem to the Identity

Let's compare the given condition \(x + y + 3 = 0\) with the condition \(a + b + c = 0\). We can see a direct correspondence if we let:

• \(a = x\)
• \(b = y\)
• \(c = 3\)

With these substitutions, the condition \(a + b + c = 0\) becomes \(x + y + 3 = 0\), which exactly matches the condition given in the problem.

Using the Identity to Simplify \(x^3 + y^3 + 3^3\)

Since \(x + y + 3 = 0\), we can use the identity \(a^3 + b^3 + c^3 = 3abc\) with \(a=x\), \(b=y\), and \(c=3\). Substituting these values into the identity, we get:

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

\(x^3 + y^3 + 27 = 9xy\)

Rearranging the Simplified Expression

We can rearrange this equation to match parts of the expression we need to evaluate (\(x^3 + y^3 - 9xy + 9\)):

\(x^3 + y^3 - 9xy + 27 = 0\)

From this, we can isolate the \(x^3 + y^3 - 9xy\) part:

\(x^3 + y^3 - 9xy = -27\)

Evaluating the Target Expression

Now we need to find the value of \(x^3 + y^3 - 9xy + 9\). We can substitute the value we just found for \(x^3 + y^3 - 9xy\):

\((x^3 + y^3 - 9xy) + 9\)

\((-27) + 9\)

\(-18\)

Conclusion

The value of the expression \(x^3 + y^3 - 9xy + 9\), given that \(x + y + 3 = 0\), is \(-18\).

Revision Table: Algebraic Identities for Exams

Understanding algebraic identities is crucial for simplifying expressions and solving equations in algebra. Here are a few common and important identities:

Additional Information: Why Algebraic Identities are Useful

Algebraic identities are equations that are true for all possible values of the variables involved. They are powerful tools in algebra because they:

• Simplify expressions: Identities help rewrite complex expressions in a simpler or more manageable form.
• Factorize polynomials: Many identities provide formulas for factoring polynomials quickly.
• Solve equations: By simplifying or rearranging expressions, identities can make solving equations easier.
• Prove other theorems: Identities form the basis for proving more complex mathematical theorems.

Recognizing patterns in algebraic expressions and matching them to known identities is a key skill in mathematics. The identity \(a^3 + b^3 + c^3 = 3abc\) when \(a+b+c=0\) is a classic example frequently used in problems involving the sum of cubes under a linear constraint.