Problem 63 - A Graphical Approach to College Algebra

Question

Factor, using the given common factor. Assume that all variables represent positive real numbers. $$9 z^{-1 / 2}+2 z^{1 / 2} ; \quad z^{-1 / 2}$$

Step-by-Step Solution

Answer

The factored form is \(z^{-1/2}(9 + 2z)\).

Step 1: Identify the Common Factor

The common factor given is \(z^{-1/2}\). Our task is to factor it out from each term in the expression: \(9 z^{-1/2} + 2 z^{1/2}\).

Step 2: Factor out the Common Factor

Factor \(z^{-1/2}\) out of each term:- For the first term, \(9 z^{-1/2}\), factoring out \(z^{-1/2}\) means dividing the term by \(z^{-1/2}\), leaving \(9\).- For the second term, \(2 z^{1/2}\), when \(z^{-1/2}\) is factored out, we divide the coefficient by \(z^{-1/2}\), resulting in \(2 (z^{1/2} / z^{-1/2}) = 2 z^{1/2 - (-1/2)} = 2z^{1} = 2z\).

Step 3: Write the Factored Expression

Combine the factored terms inside a parenthesis and multiply by the common factor, \(z^{-1/2}\):\(z^{-1/2}(9 + 2z)\).

Step 4: Verification

Verify the factorization by distributing \(z^{-1/2}\) back into the expression:- \(z^{-1/2} \times 9 = 9z^{-1/2}\)- \(z^{-1/2} \times 2z = 2z^{1/2}\)The original expression \(9 z^{-1/2} + 2 z^{1/2}\) is recovered, confirming that the factoring is correct.

Key Concepts

Common Factor

In algebra, identifying a common factor is the first step in simplifying expressions through factoring. A common factor is a term or number that divides each term in the expression without leaving a remainder. In the exercise provided, the common factor is given as \(z^{-1/2}\). This means it is the term that appears in both parts of the expression, \(9z^{-1/2} + 2z^{1/2}\).

• Look at the complete expression and pinpoint the term that can be consistently factored out.
• Each term should be able to be divided by this common factor.

In this example, factor \(z^{-1/2}\) is identified because it is shared across the terms, making it possible to factor out to simplify the expression.

Exponent Rules

Exponent rules are crucial when dealing with expressions involving powers of variables. Understanding these rules helps simplify expressions and factor them accurately. In the exercise, exponent rules are applied as follows:

• When dividing expressions with the same base, subtract their exponents: \(z^{a} / z^{b} = z^{a - b}\).
• This is how the expression \(2 z^{1/2} / z^{-1/2}\) simplifies to \(2z^{1 - (-1/2)} = 2z^{1/2 + 1/2} = 2z^{1} = 2z\).

These exponent rules are key to manipulating terms so they can be added, subtracted, multiplied, or divided properly during factoring.

Factoring step-by-step

Factoring step-by-step involves systematically simplifying an expression by pulling out the common factor. Here’s how it works using the step-by-step approach from the original solution:1. **Identify the Common Factor:** Begin by recognizing \(z^{-1/2}\) as the shared factor in each term of the expression.2. **Factor it out:** Rewrite each term by dividing it by the common factor. - For \(9 z^{-1/2}\), division by \(z^{-1/2}\) leaves \(9\). - For \(2 z^{1/2}\), dividing by \(z^{-1/2}\) gives \(2z\) after adjusting exponents via the rules.3. **Rewrite the Expression:** Combine terms inside a parenthesis and multiply by the common factor, getting \(z^{-1/2} (9 + 2z)\).4. **Verification:** Ensure accuracy by multiplying back through the parentheses: - \(z^{-1/2} \times 9 = 9z^{-1/2}\) - \(z^{-1/2} \times 2z = 2z^{1/2}\)It confirms that \(z^{-1/2} (9 + 2z)\) is a correct factorization as we recapture the original expression: \(9 z^{-1/2} + 2 z^{1/2}\).