Problem 86 - Algebra and Trigonometry

Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ (x-1)^{7 / 2}-(x-1)^{3 / 2} $$

Step-by-Step Solution

Answer

The expression factors as \((x-1)^{3/2}(x-2)(x)\).

Step 1: Identify Common Factors

The expression is \((x-1)^{7/2} - (x-1)^{3/2}\). Identify the common factor: both terms have \((x-1)\) raised to a power, \((x-1)^{3/2}\) is the common factor.

Step 2: Factor Out the Common Factor

Factor out the smallest power of \((x-1)\), which is \((x-1)^{3/2}\). Write the expression as \[(x-1)^{3/2}((x-1)^{7/2 - 3/2} - 1)\].

Step 3: Simplify the Expression

Simplify the expression inside the parenthesis: \((x-1)^{7/2 - 3/2} - 1\) becomes \((x-1)^{4/2} - 1\), which simplifies further to \((x-1)^2 - 1\).

Step 4: Recognize and Apply the Difference of Squares

The expression \((x-1)^2 - 1\) is a difference of squares: \((a)^2 - (b)^2 = (a-b)(a+b)\) where \(a = (x-1)\) and \(b = 1\). Factor it as \([(x-1) - 1][(x-1) + 1]\) which simplifies to \((x-2)(x)\).

Step 5: Write the Complete Factored Form

Combine all parts to write the complete factored expression: \((x-1)^{3/2}((x-2)(x))\). Thus, the expression \((x-1)^{7/2} - (x-1)^{3/2}\) factors to \((x-1)^{3/2}(x-2)(x)\).

Key Concepts

Common Factors

When tackling the problem \((x-1)^{7/2}-(x-1)^{3/2}\), our very first step is to hunt for common factors. In mathematics, a common factor is a term that is found in every part of an expression. Basically, it's a piece that you can "pull out" to make things simpler. Here, both parts of the expression share a little something: \((x-1)\). Not just \((x-1)\) itself, but raised to a power. The secret sauce is to target the smallest power of that common factor, which is \((x-1)^{3/2}\), and use it for simplifying the expression. By doing this, you create a smaller, more manageable expression that opens the gates for further simplification.

Difference of Squares

In mathematics, you'll frequently come across the difference of squares. It's a special pattern and a real handy tool when factoring. The general idea is about expressions like \(a^2 - b^2\), which you can crack open as \((a - b)(a + b)\). This pattern is incredibly useful and speeds up factoring.
For our given problem, after isolating the common factor, we ended up with something that resembles a difference of squares: \((x-1)^2 - 1\). Here, \(a = (x-1)\) and \(b = 1\). Using the difference of squares idea lets us simplify the expression further into: \([(x-1) - 1][(x-1) + 1]\), which neatly turns into \((x-2)(x)\). This transformation helps in arriving at a neatly factored form by breaking down the expression algebraically.

Exponents

Exponents, also known as powers, are a shorthand way to express repeated multiplication. In the expression \((x-1)^{7/2}-(x-1)^{3/2}\), exponents are at the heart of the problem.
To untangle this expression, we leverage rules about exponents which state you can subtract powers when dividing similar bases. This is why when isolating the common factor \((x-1)^{3/2}\), what follows inside the parentheses is simply the exponent difference:\((x-1)^{7/2 - 3/2} - 1\). This becomes \((x-1)^{4/2} - 1\), or \((x-1)^2 - 1\) when simplified. Understanding how to work with exponents is key to streamlining complex expressions into simpler forms.

Simplification

Simplification is the process of making an expression more compact and easier to understand. In our exercise, simplification happens at several steps and is vital in breaking down the expression neatly.
Once the common factor is factored out, you must simplify what's left in the parentheses: \((x-1)^{7/2 - 3/2} - 1\), which narrows down to \((x-1)^2 - 1\). Recognizing this as a difference of squares allows us to further simplify to the factors \((x-2)(x)\). Through this process, simplification takes a complex piece and makes it understandable, applying patterns like difference of squares and basic operations on exponents to achieve a truly factored result. It's like cleaning up clutter—making sure everything is in its neatest form.