Question

Describe and correct the error in factoring the polynomial. $$ \begin{aligned} x^9 &+8 x^3=\left(x^3\right)^3+(2 x)^3 \\ &=\left(x^3+2 x\right)\left[\left(x^3\right)^2-\left(x^3\right)(2 x)+(2 x)^2\right] \\ &=\left(x^3+2 x\right)\left(x^6-2 x^4+4 x^2\right) \end{aligned} $$

Short answer

The error in the initial factoring was in the usage of the formula for sum of cubes. The corrected factorization of the given polynomial is \((x^3 + 2x)(x^6 - 2x^4 + 4x^2)\)

Step-by-step solution

Evaluate the Initial Factoring

Looking at the provided solution, we notice that the student started right by recognizing a sum of cubes. The essence of that approach is to rewrite the expression in the form \((a^3+b^3)\) where \(a=x^3\) and \(b=2x\). However, while applying the sum of cubes formula which is given by \((a^3 + b^3) = (a + b)(a^2 - ab + b^2)\), the student made an error in the second bracket, it should have been \((a - b)\) instead of \((a + b)\). Let's correct that.

Correct the Factoring

Applying the correct formula gives us: \(x^9 + 8x^3 = (x^3)^3 + (2x)^3\) = \((x^3 + 2x)((x^3)^2 - (x^3)(2x) + (2x)^2)\). Therefore, we find that the corrected factorization of the given polynomial is \((x^3 + 2x)(x^6 - 2x^4 + 4x^2)\).

Review the solution

Always remember to verify your solution by multiplying the factors back together and confirming that the result matches the original equation. If it doesn't match, then an error was made during the process. In this case, the right factorization indeed leads us back to the original equation. Thus, the solution is correct.

Key concepts

Sum of Cubes

Understanding the sum of cubes formula is pivotal to factoring certain types of polynomials efficiently. The sum of cubes formula states that for any two terms, 'a' and 'b', the expression \(a^3 + b^3\) can be factored into the product of a binomial \(a + b\) and a trinomial \(a^2 - ab + b^2\).

Let's break it down with an example. If we have the expression \(x^3 + 27\), we identify \(a = x\) and \(b = 3\), because \(3^3 = 27\). Using the sum of cubes formula, we factor this as \[x^3 + 3^3 = (x + 3)(x^2 - 3x + 9)\].

This factoring technique is not only a shortcut but also a necessary tool for solving higher-level polynomial equations. It allows us to transform an otherwise complex cubic equation into a more manageable form that can be further analyzed or solved.

Polynomial Factorization Errors

A common issue when factoring polynomials is making algebraic errors, which can alter the outcome of a problem. It's essential to apply formulas and properties accurately. In the given exercise, the error occurred while applying the sum of cubes formula. The mistake was in the signs inside the trinomial. Instead of the correct pattern of \(a^2 - ab + b^2\), the student used \(a^2 - a \cdot b + b^2\), where the sign between \(a \cdot b\) and \(b^2\) should have been positive.

To avoid errors:

• Check the signs in each term carefully.
• Ensure proper alignment of terms when writing and simplifying expressions.
• Always double-check work by re-multiplying the factors to see if the original expression is obtained.

Committing to these practices reduces the chance of error and increases the likelihood of correctly solving polynomial factorization problems.

Algebra 2 Problem Solving

Algebra 2 problem solving often involves recognizing patterns and applying appropriate formulas or theorems. The ability to identify the type of polynomial and factor it correctly is a crucial skill. The exercise demonstrates the factoring of a sum of cubes, which is a specific type of problem in this field.

To master Algebra 2 problems:

• Understand the core concepts like polynomial classifications, factorization techniques, and equations.
• Practice applying these concepts to various problems to build skill and confidence.
• Remember that verifying solutions is as important as finding them. Always check your work.

These steps help ensure that you not only arrive at the correct answer but also solidify your understanding of the underlying algebraic principles.