Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. \(F^{3}+L^{3}=(\quad+\quad)\left(F^{2}-F L+L^{2}\right)\) b. \(F^{3}-L^{3}=(F \quad L)(\quad+F L+\quad)\)

Step-by-Step Solution

Answer

a. \( F + L \) b. \( F - L, \ F^2 + FL + L^2 \)

Step 1: Identify Formula Structure

The expressions given in the question resemble the formulas for the sum and difference of cubes. For sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). For difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Let's use these formulas in the given expressions.

Step 2: Solve Part (a)

Given expression: \(F^3 + L^3 = ( ext{blank} + ext{blank})(F^2 - FL + L^2)\). By comparing this to the formula for sums of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \), it is clear that the blanks represent \(F + L\). Therefore, \(F^3 + L^3 = (F + L)(F^2 - FL + L^2)\).

Step 3: Solve Part (b)

Given expression: \(F^3 - L^3 = (F ext{ blank } L)( ext{blank} + FL + ext{blank})\). This matches the difference of cubes formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Thus, the blanks represent \(F\), \(L\), and \(L^2\) respectively. So, \(F^3 - L^3 = (F - L)(F^2 + FL + L^2)\).

Key Concepts

Sum of Cubes

The sum of cubes is a formula used to factor expressions of the form \( a^3 + b^3 \). This might look a bit intimidating at first, but the formula makes it easier:\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]Why is this helpful? When faced with an expression like \( F^3 + L^3 \), using the formula allows us to break it down. The first term \( (a + b) \) simply adds the cube roots of the terms, and the second term \( (a^2 - ab + b^2) \) puts together their squares and the product with a minus. This formula provides a neat factorization of the sum of two cubes—transforming it into a straight multiplication that joins polynomials together.

Difference of Cubes

The difference of cubes is another handy formula for factoring expressions that look like \( a^3 - b^3 \). Again, the formula is here to save the day:\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]If we break it down:

• The first factor \( (a - b) \) is just subtracting one cube root from the other.
• The second factor \( (a^2 + ab + b^2) \) organizes the squares of the terms and the product with a plus this time.

This method is particularly useful for simplifying expressions such as \( F^3 - L^3 \). By doing so, it's turned into something simpler: a difference multiplied by a trinomial, making it easier to work with.

Factoring

Factoring is a cornerstone concept in algebra, a process that splits an expression into products of simpler ones. In our exercises, factoring focuses on special polynomial identities—the sum and difference of cubes. To factor efficiently, apply:

• The sum of cubes formula whenever you see the sum of two cubes.
• The difference of cubes formula for the difference of two cubes.

These predetermined formulas take the guesswork out of factoring these specific types of polynomials. Not only do they simplify expressions by turning them into more manageable pieces, but they also offer neat and tidy sets of multipliers that make further algebraic manipulations much clearer. Factoring thus becomes an essential part of solving complex equations and expressions effectively.