Problem 80 - Algebra and Trigonometry

Question

77–84 ? 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

\((x-1)^{3/2}(x)(x-2)\) is the completely factored form.

Step 1: Identify the Common Factor

The given expression is \((x-1)^{7/2}-(x-1)^{3/2}\). We need to factor out the lowest power of \((x-1)\) from each term. The lowest power between \(7/2\) and \(3/2\) is \(3/2\).

Step 2: Factor Out the Common Factor

Factor \((x-1)^{3/2}\) out of the expression:\[(x-1)^{7/2} - (x-1)^{3/2} = (x-1)^{3/2} \,\left((x-1)^2 - 1\right).\]

Step 3: Further Simplify the Remaining Expression

The expression \((x-1)^2 - 1\) is a difference of squares. Recall the formula for the difference of squares: \(a^2 - b^2 = (a-b)(a+b)\). Letting \(a = (x-1)\) and \(b = 1\), we can rewrite the expression as:\[((x-1) - 1)((x-1) + 1).\]

Step 4: Simplify the Difference of Squares

Simplify the expression obtained from the difference of squares:\[(x-2)(x).\]

Step 5: Write the Complete Factored Form

Substituting back, the completely factored form of the original expression is:\[(x-1)^{3/2}(x)(x-2).\]

Key Concepts

Common Factor

When tackling polynomial expressions, one of the first steps in simplifying or factoring is identifying the common factor. This helps streamline the expression and prepares it for further simplification. To find the common factor in an expression like \((x-1)^{7/2} - (x-1)^{3/2}\), you need to look for the lowest power of a repeated base, in this case \((x-1)\). Here, the exponents are \(7/2\) and \(3/2\), so the smallest one is \(3/2\).

By factoring out \((x-1)^{3/2}\), you essentially remove or 'pull' this factor out of each term in the polynomial:
\[ (x-1)^{7/2} - (x-1)^{3/2} = (x-1)^{3/2}((x-1)^2 - 1). \]

This method simplifies working with the expression by reducing the powers of the original components and revealing further relationships within the polynomials.

Difference of Squares

Once you have factored out the common factor, you often find expressions that can be simplified further. In our case, we end up with \((x-1)^2 - 1\). This particular form is a "difference of squares", a shortcut or identity useful for fast simplification.

A difference of squares applies the formula:
\[ a^2 - b^2 = (a-b)(a+b). \]
This formula can be used because the term \((x-1)^2\) is a perfect square, as is \(1\). By identifying this, you can rewrite it as:

• First, assign \(a = (x-1)\)
• Then assign \(b = 1\)
• Apply the formula:

\[ ((x-1) - 1)((x-1) + 1) = (x-2)(x). \]

Recognizing a difference of squares in your initial polynomial can be crucial for efficient simplification and achieving your completely factored form.

Simplification Process

The simplification process in polynomial factoring often involves systematically using algebraic identities and techniques like factoring common factors and recognizing patterns such as the difference of squares. The solution involves breaking down complex expressions into simpler parts for easier manipulation.

In our example, after factoring out the common power, the expression reduced to \((x-1)^{3/2}((x-1)^2 - 1)\). The simplification continued by acknowledging \((x-1)^2 - 1\) as a difference of squares, further refining the expression to \((x-2)(x)\).

Finally, when all factors are clearly separated and no further simplification is possible, you compile the fully factored expression:
\[ (x-1)^{3/2}(x)(x-2). \]

Understanding and applying these simplification steps help transform superficially complex expressions into a manageable product of terms, highlighting the power and simplicity of algebra. This fosters clearer solutions and enhances problem-solving efficiency.