Problem 67 - A Graphical Approach to College Algebra

Question

Factor, using the given common factor. Assume that all variables represent positive real numbers. $$(p+4)^{-3 / 2}+(p+4)^{-1 / 2}+(p+4)^{1 / 2} ; \quad(p+4)^{-3 / 2}$$

Step-by-Step Solution

Answer

The expression factors to \((p+4)^{-3/2} \cdot (1 + (p+4) + (p+4)^2)\).

Step 1: Identify the Common Factor

The expression given is \((p+4)^{-3/2} + (p+4)^{-1/2} + (p+4)^{1/2}\). We are told to factor out \((p+4)^{-3/2}\), which is the smallest power of \((p+4)\) in the expression. In mathematical terms, this means expressing each term in the sum in terms of \((p+4)^{-3/2}\).

Step 2: Rewrite Each Term to Factor Out the Common Factor

Reformulate each term in the expression by factoring out \((p+4)^{-3/2}\). - The first term \((p+4)^{-3/2}\) remains as it is upon factoring: \(1 \times (p+4)^{-3/2}\).- The second term \((p+4)^{-1/2}\) can be rewritten by dividing its power by the common factor: \((p+4)^{-1/2} = (p+4)^{-3/2} \cdot (p+4)\).- The third term \((p+4)^{1/2}\) can be rewritten similarly: \((p+4)^{1/2} = (p+4)^{-3/2} \cdot (p+4)^2\).

Step 3: Express the Entire Expression with the Common Factor

Now that we have rewritten each term, we combine them into a single expression:\[(p+4)^{-3/2} \times (1 + (p+4) + (p+4)^2)\]. This showcases the common factor \((p+4)^{-3/2}\) with a quadratic expression within the parentheses.

Step 4: Final Verification

Check the factorization by distributing the common factor back in:\((p+4)^{-3/2} \cdot 1 = (p+4)^{-3/2}\),\((p+4)^{-3/2} \cdot (p+4) = (p+4)^{-1/2}\),\((p+4)^{-3/2} \cdot (p+4)^2 = (p+4)^{1/2}\).All terms return to their original form, confirming the factorization is correct.

Key Concepts

Common Factor

A common factor is an element that exists equally well across multiple terms within an algebraic expression. In our exercise, we're dealing with terms that involve different powers of \((p+4)\). The goal is to factor out the smallest power present in these terms, which simplifies the expression.
Consider the steps involved:

• Identify the smallest power of the group, in this case, \((p+4)^{-3/2}\).
• Rewrite each term by dividing its power by this smallest power.

This "smallest power" acts as a common factor, making it easier to manage and simplify complex algebraic expressions. By factoring out the common factor, you simplify what used to be a combination of different powers into a singular, more manageable unit.

Exponent Rules

Exponent rules are fundamental in simplifying algebraic expressions, especially when multiple terms involve the same base. Recall some basic rules:

• Multiplying powers of the same base: \(a^m \cdot a^n = a^{m+n}\)
• Dividing powers of the same base: \(a^m \div a^n = a^{m-n}\)
• Raising a power to another power: \((a^m)^n = a^{m \cdot n}\)

In our given expression, we used these exponent rules to simplify each term so that they all include the common factor of \((p+4)^{-3/2}\).For example, when rewriting \((p+4)^{-1/2}\) as \((p+4)^{-3/2} \cdot (p+4)\), you use the division rule (subtract exponents) to adjust the powers to match the factored form of the smallest power, i.e., the common factor. This process emphasizes why mastering exponent rules is critical for factoring and simplifying expressions.

Quadratic Expression

After factoring out the common factor \((p+4)^{-3/2}\), the remaining expression inside the parenthesis became quadratic: \((1 + (p+4) + (p+4)^2)\).A quadratic expression typically follows the structure \(ax^2 + bx + c\). Although in this case, it reveals a more complex structure because it integrates the term \((p+4)\), it manifests as a quadratic due to its second degree.
Understanding quadratic expressions:

• They appear frequently in algebra, offering solutions through various methods like factoring, completing the square, and the quadratic formula.
• Recognizing the format helps in predicting the behavior of expressions and equations.

This quadratic structure is instrumental in forecasting a possible parabolic shape graphically, and it assists in further operations such as integration and differentiation if continued into calculus.