Problem 18 - Algebra 2

Question

Simplify. $$ \left(x^{3}-27\right) \div(x-3) $$

Step-by-Step Solution

Answer

\(x^2 + 3x + 9\)

Step 1: Recognize the Polynomial Identity

Notice that the expression \(x^3 - 27\) is a difference of cubes. Recall the formula for factoring a difference of cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). In this case, \(a = x\) and \(b = 3\).

Step 2: Apply the Difference of Cubes Formula

Using the identity from Step 1, factor \(x^3 - 27\) into \((x-3)(x^2 + 3x + 9)\). Thus, the expression becomes \((x-3)(x^2 + 3x + 9) \div (x-3)\).

Step 3: Simplify the Expression

Since \((x-3)\) is a common factor in both the numerator and the divisor, you can cancel it out. You're left with the simplified expression \(x^2 + 3x + 9\).

Step 4: Verify the Result

Finally, verify the result by substituting a sample value for \(x\) in both the original and simplified expressions to ensure they yield the same value. This confirms the simplification is correct.

Key Concepts

Difference of Cubes

In algebra, a "difference of cubes" refers to an expression where two cubed terms are subtracted. The general formula for factoring a difference of cubes is given by \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). This formula is a powerful tool for simplifying such expressions.Consider the expression \(x^3 - 27\). This is a classic example of a difference of cubes, where \(a = x\) and \(b = 3\) (since \(27 = 3^3\)). Applying the difference of cubes formula helps us rewrite the expression as \((x-3)(x^2 + 3x + 9)\).

• Quick Tip: Identifying the terms \(a\) and \(b\) is crucial for applying the formula correctly.
• Remember: The middle term \(ab\) in \((a^2 + ab + b^2)\) often helps consolidate understanding.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler, multiplicative expressions known as factors. These factors, when multiplied together, yield the original polynomial.In the context of our example, we factored \(x^3 - 27\) into \((x-3)(x^2 + 3x + 9)\). The first factor, \(x-3\), corresponds to one of the simple roots of the polynomial, while the trinomial \(x^2 + 3x + 9\) completes the factorization.

• Helpful Hint: Always look for common identities like difference or sum of cubes to make factoring easier.
• Goal: Factoring simplifies the polynomial, making it easier to divide or solve.

Simplifying Expressions

Simplifying an expression means reducing it to its simplest form. This often involves canceling common factors, combining like terms, or employing algebraic identities.In our example, after factoring \(x^3 - 27\), the expression becomes \((x-3)(x^2 + 3x + 9) \div (x-3)\). With \(x-3\) appearing in both the numerator and the divisor, it can be cancelled out. This leaves us with the simplified form \(x^2 + 3x + 9\).

• Important Note: Cancel only common factors present across the entire numerator or denominator.
• Verification: Checking your final expression with sample values ensures the simplification is legitimate.