Problem 28 - Algebra for College Students

Question

Perform the indicated divisions. $$ \frac{x^{3}+64}{x+4} $$

Step-by-Step Solution

Answer

The result of the division is \( x^2 - 4x + 16 \).

Step 1: Identify the Division

The given division problem is to divide the polynomial \( x^3 + 64 \) by \( x + 4 \). This is a polynomial division problem where \( x^3 + 64 \) is the dividend and \( x + 4 \) is the divisor.

Step 2: Factor the Dividend

Notice that \( x^3 + 64 \) is a sum of cubes. We can factor it into \((x+4)(x^2-4x+16)\) using the sum of cubes formula: \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \), where \( a = x \) and \( b = 4 \).

Step 3: Simplify the Division

Now, the expression \( \frac{x^3+64}{x+4} \) becomes \( \frac{(x+4)(x^2-4x+16)}{x+4} \). We can simplify this by canceling out the common factor \( x+4 \) in the numerator and the denominator.

Step 4: Write the Final Result

After canceling the common factor, the simplified expression is \( x^2 - 4x + 16 \). This is the quotient of the division. Hence, \( \frac{x^3+64}{x+4} = x^2 - 4x + 16 \).

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is an important identity in algebra that helps to factor expressions of the form \(a^3 + b^3\). The formula is given by:\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \] This formula is particularly useful when dealing with polynomial expressions that resemble a cubic sum. In the given exercise, \(x^3 + 64\) is a classic example of a sum of cubes, where \(a = x\) and \(b = 4\). Using the formula, \(x^3 + 64\) can be factored into

• \((x+4)\)
• \(x^2 - 4x + 16\)

This step is crucial because it simplifies the process of division by breaking down polynomials into more manageable parts. Remember that correctly identifying the structure as a sum of cubes is key to applying this formula effectively, making polynomial division straightforward.

Simplifying Rational Expressions

Simplifying rational expressions is all about reducing fractions to their simplest form. Just as we simplify numerical fractions by canceling out common factors, the same principle applies to algebraic fractions. In the exercise, we need to simplify the expression:\[ \frac{(x+4)(x^2-4x+16)}{x+4} \] Here, the common factor \((x+4)\) appears in both the numerator and the denominator. When this happens, we can cancel these common terms, much like we would cancel factors in the fraction \(\frac{6}{6}\) resulting in 1. This cancellation leaves us with:

• \(x^2 - 4x + 16\)

It's important to recognize common factors and apply cancelation accurately, as it simplifies expressions efficiently. This technique not only reduces the complexity of expressions but also aids in solving equations and understanding algebraic concepts better. Keep in mind that this step is only valid if the factor is not zero, which means when \(x eq -4\) in this case.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler 'factors' that, when multiplied, give the original polynomial. It's a foundational skill in algebra, serving multiple purposes like simplifying expressions and solving polynomial equations. Let's consider the polynomial from the problem: \(x^3 + 64\). To factor it fully: 1. Identify the pattern or special formula applicable. In this case, it is the sum of cubes.2. Use the sum of cubes formula mentioned earlier to factor \(x^3 + 64\) into \((x+4)(x^2-4x+16)\). Factoring is the first critical step in simplifying expressions like the one in this problem. Recognizing such patterns helps in solving algebraic problems more efficiently. It also improves the understanding of how different algebraic identities work, laying the foundation for more advanced topics. Whether simplifying expressions or solving complex equations, learning to factor polynomials effectively is invaluable.