Problem 13 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{a^{3}+b^{3}}{a+b} $$

Step-by-Step Solution

Answer

The simplified expression is \(a^2 - ab + b^2\).

Step 1: Recognize the Formula

Recognize that the expression \(a^3 + b^3\) can be simplified using the formula for the sum of cubes, which is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).

Step 2: Substitute the Formula

Substitute \(a^3 + b^3\) with \((a + b)(a^2 - ab + b^2)\) in the given expression to get:\(\frac{(a + b)(a^2 - ab + b^2)}{a + b}\)

Step 3: Simplify by Canceling

Since \(a + b\) is present in both the numerator and the denominator, you can cancel them out.The expression simplifies to:\(a^2 - ab + b^2\)

Step 4: Write the Final Expression

The simplified expression is \(a^2 - ab + b^2\). This is the answer after simplifying the given expression using the formula for the sum of cubes.

Key Concepts

Sum of Cubes

The sum of cubes is a special algebraic expression that takes the form \( a^3 + b^3 \). It is important to recognize this form because it can be factored using a specific formula. Understanding this can greatly simplify your work, just like it does for the original exercise.To factor the sum of cubes, use the formula: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]Here's a breakdown of what each component represents:

• \( (a + b)\) is a simple sum of the bases \(a\) and \(b\).
• \( (a^2 - ab + b^2)\) is a trinomial that comprises the squares and the product of these bases.

When you encounter \( a^3 + b^3 \), applying this formula allows leveraging their sum to further simplify expressions, which is exactly what we do in our example.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It's used to break down complex expressions into simpler parts. In our exercise, factoring is the key technique used to dismantle \( a^3 + b^3 \).We employ the sum of cubes formula from the previous section. It helps to express \( a^3 + b^3 \) in terms of \( (a + b)\) and \( (a^2 - ab + b^2)\). This process involves writing a polynomial as a product of its simpler factors. If you have these factors, it often becomes easier to solve or simplify algebraic expressions.For example:

• Recognize polynomial forms like sum of cubes, difference of cubes, or quadratic trinomials.
• Apply the corresponding formulas to factor them systematically.

By becoming comfortable with factoring, you unlock the ability to manipulate and simplify expressions even more effectively.

Simplification Techniques

Simplification is about reducing expressions into their most basic form. It's a crucial skill in algebra, as it makes problems easier to tackle. In our problem, simplification involves a few different strategies.First, we use the sum of cubes formula, identifying the expression correctly and substituting it with its factored equivalent: \( \frac{(a + b)(a^2 - ab + b^2)}{a + b} \)Next, we simplify by canceling out terms that appear in both the numerator and the denominator. In our case, \(a + b\) is present in both, allowing us to remove it:

• Cancel common factors when both numerator and denominator share them.

Finally, the expression is left in a much simpler form: \(a^2 - ab + b^2\)These simplification strategies help streamline calculations, leading to quicker and often more elegant solutions.