Question

Sum or Difference of Two Cubes. $$a^{3}-343$$

Short answer

\((a - 7)(a^2 + 7a + 49)\)

Step-by-step solution

Identify the Difference of Cubes Formula

Recognize the expression as a difference of cubes which can be factored using the formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Determine the values of \(a\) and \(b\)

From the given expression \(a^{3}-343\), we can see that \(a^3\) is the cube of \(a\) and \(343\) is the cube of \(7\) because \(7\times7\times7=343\). So, \(a = a\) and \(b = 7\).

Apply the Difference of Cubes Formula

Substitute \(a\) and \(b\) into the formula to get: \((a - 7)(a^2 + 7a + 49)\).

Write the Final Factored Form

The expression \(a^{3}-343\) is now factored as \((a - 7)(a^2 + 7a + 49)\).

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into simpler components (or factors) that, when multiplied together, give back the original polynomial. It's similar to finding the ingredients that make up your favorite recipe. Just like how a recipe is created using a combination of ingredients, a polynomial is made of the product of its factors.

For example, when faced with a polynomial expression like the difference of two cubes, \(a^3 - 343\), we start by looking for expressions that could be the 'building blocks' of this polynomial. Recognizing that 343 is a cube number, we identify that we are dealing with a special factoring case. Following the appropriate formula, the polynomial \(a^3 - 343\) can be decomposed into \(a - 7\) and \(a^2 + 7a + 49\). Each piece of this decomposition is a polynomial in its own right but in a much simpler form. Factoring polynomials helps make complex algebraic calculations and simplifications more manageable.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and embody the relationships between quantities using variables, numbers, and arithmetic operations. Think of them as sentences that describe how quantities interact, with the variables acting like pronouns taking the place of unknown values. These expressions can take many forms, from simple like \(2x + 3\) to more complex ones like the difference of cubes \(a^3 - b^3\).

The ability to manipulate and rewrite algebraic expressions is akin to having a conversation in a foreign language. It involves understanding the grammar (in this case, rules of algebra) and vocabulary (operators, constants, and variables). For instance, \(a^3 - 343\) can be whispered in algebraic terms as \(a - 7)(a^2 + 7a + 49)\), where every element plays a part in the narrative of the expression. Engaging with and mastering the nuances of algebraic expressions allows students to unlock the meaning behind the numbers and symbols, transforming abstract concepts into tangible solutions.

Exponentiation

Exponentiation is the operation of raising one quantity to the power of another. In simpler terms, it's like taking a number and multiplying it by itself a certain number of times, as specified by the exponent. For example, the notation \(7^3\) means multiplying 7 by itself three times (\(7 \times 7 \times 7\)), which equals 343.

Understanding exponentiation is crucial when dealing with polynomial expressions, especially when we encounter terms like \(a^3\) or \(b^3\). These 'cubed' terms suggest a three-dimensional expansion of a quantity and play a special role in patterns such as the difference of cubes. Exponentiation isn't just about multiplication on steroids; it's a powerful concept that aids in modeling real-world situations where growth or decay happens rapidly, such as in compound interest or certain natural processes. Mastering exponentiation can help students to not only solve problems more efficiently but also to appreciate the beauty and utility of mathematics in both theoretical and practical scenarios.