Problem 94 - Elementary Algebra

Question

(a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use a pattern to factor \(p^{3}+8 w^{3}\). Incorrect Answer: Since the pattern is \(a^{3}+b^{3}=\) \((a+b)\left(a^{2}-a b+b^{2}\right), a=p\) and \(b=8 w\), and the factored polynomial is \((p+8 w)\left(p^{2}-8 p w+64 w^{2}\right)\).

Step-by-Step Solution

Answer

\( (p + 2w)(p^2 - 2pw + 4w^2) \)

Step 1: Identify the mistake

The error lies in the incorrect substitution of the values for the variables in the pattern. The formula for factoring the sum of cubes is correctly stated as \[a^{3}+b^{3}= (a+b)(a^{2}-ab+b^{2})\]However, in this problem, the terms were substituted incorrectly. Specifically, the term for \(b\) should be \(2w\), not \(8w\).

Step 2: Re-identify the pattern terms

Rewrite the given expression \(p^{3}+8w^{3}\) in the form of \(a^{3}+b^{3}\). Notice that \[8w^{3} = (2w)^{3}\]Thus, \(a = p\) and \(b = 2w\).

Step 3: Apply the pattern correctly

Using the correct values for \(a\) and \(b\), substitute them into the sum of cubes pattern: \[(a + b)(a^2 - ab + b^2)\]Substitute \(a = p\) and \(b = 2w\) into the formula:\[(p + 2w)(p^2 - p(2w) + (2w)^2)\]

Step 4: Simplify the expression

Simplify each term inside the parentheses: \[ (p + 2w) (p^2 - 2pw + 4w^2) \]

Step 5: Final answer

The correctly factored expression is\[ (p + 2w)(p^2 - 2pw + 4w^2) \]

Key Concepts

Sum of Cubes

To solve the polynomial expression given in the problem, we use the concept of the sum of cubes. The sum of cubes formula is: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\) This formula helps factorize the polynomial into two smaller expressions.

In the problem, we have the expression \(p^{3} + 8w^{3}\). To use the sum of cubes formula, we need to recognize that \(8w^{3} = (2w)^{3}\), making our \(a = p\) and \(b = 2w.\) By plugging these values into the sum of cubes formula, which should yield the correctly factored form. Understanding and correctly applying this formula is essential, as it simplifies problems that involve cubic polynomials significantly.

Algebraic Patterns

Algebraic patterns, like the sum of cubes, are crucial for simplifying and solving polynomial expressions. These patterns are essentially shortcuts that can save time and effort in problem-solving.

One common algebraic pattern besides the sum of cubes is the difference of squares, given by: \(a^{2} - b^{2} = (a + b)(a - b)\).

Mastering these patterns will greatly benefit you as they frequently appear in algebra. They enable quick factoring of expressions that would otherwise be more complex to solve. Practicing these patterns, recognizing them in various forms of equations, and correctly substituting values can vastly improve your algebra skills.

Polynomial Factorization

Factoring polynomials is a fundamental skill in algebra. It involves breaking down a complex polynomial into simpler factors that, when multiplied together, give the original polynomial. This process is crucial for solving equations, simplifying expressions, and analyzing functions.

For the sum of cubes: \(p^{3} + 8w^{3}\), proper identification and correct substitution of the terms are critical. Recognizing that \(8w^{3} = (2w)^{3}\) allows us to rewrite the expression in a recognizable form for factorization.

The correct factored form for \(p^{3} + 8w^{3}\) is determined by the sum of cubes formula: \( (p + 2w)(p^2 - 2pw + 4w^2)\). Frequently practicing polynomial factorization helps in understanding the structure of these equations and improves problem-solving efficiency.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations (like addition, subtraction, multiplication, and division). They are the foundation of algebra and serve as a stepping stone for more complex mathematical concepts.

In our problem, the expression \(p^{3} + 8w^{3}\) is an algebraic expression that requires factoring for simplification. Understanding how to manipulate and transform these expressions using algebraic rules and patterns is critical.

For example, knowing how to rewrite \(8w^{3}\) as \( (2w)^{3}\) enables the use of the sum of cubes formula. Breaking down the expression into manageable parts (\[ (p + 2w)(p^2 - 2pw + 4w^2) \]) shows the power of simplifying algebraic expressions correctly. With a firm grasp on algebraic expressions and patterns, one can seamlessly navigate through various algebra problems.