Problem 44 - College Algebra

Question

For the following exercises, factor the polynomials. $$ 4 x(x-1)^{-\frac{2}{3}}+3(x-1)^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The factored form is \((x-1)^{-\frac{2}{3}}(7x - 3)\)."

Step 1: Identify Common Factors

To factor the polynomial \( 4x(x-1)^{-\frac{2}{3}} + 3(x-1)^{\frac{1}{3}} \), start by looking for any common factors in both terms. We see that both terms include \((x-1)\) raised to some power. The common factor here is \((x-1)^{-\frac{2}{3}}\), which is the smaller of the two exponents across the terms.

Step 2: Factor Out the Common Element

Factor \((x-1)^{-\frac{2}{3}}\) out of each term:\[ 4x(x-1)^{-\frac{2}{3}} + 3(x-1)^{\frac{1}{3}} = (x-1)^{-\frac{2}{3}} \left[ 4x + 3(x-1)^{\frac{1}{3} - (-\frac{2}{3})} \right] \]Simplifying the exponent in the bracket gives:\[ 3(x-1)^{\frac{1}{3} + \frac{2}{3}} = 3(x-1)^{1} \] Thus, we have:\[(x-1)^{-\frac{2}{3}} \left( 4x + 3(x-1) \right)\].

Step 3: Simplify the Expression

Now simplify the expression inside the brackets:\[4x + 3(x-1) = 4x + 3x - 3 = 7x - 3\]. This gives us the fully factored form:\[(x-1)^{-\frac{2}{3}}(7x - 3)\].

Key Concepts

Common Factors

When working with polynomials, identifying common factors is an important first step toward factoring. A common factor is a term that appears in all terms of the polynomial expression. By factoring out common elements, expressions become simpler and easier to manage. In the case of the exercise provided, both terms in the polynomial include a factor of \((x-1)\). We look for the smallest power of this common factor, which is \((x-1)^{-\frac{2}{3}}\) in our exercise. By identifying and factoring out common factors, we lay the groundwork for further simplification of the polynomial expression.

• Identify the common factor in each term.
• Ensure you choose the smallest exponent of the common factor for consistency.
• Factor it out, reducing each term by the common factor identified.

Exponent Rules

Exponent rules are essential when factoring polynomials as they provide the basis for understanding how terms can be manipulated. When dealing with powers, substitutions, and simplification, the following rules often come into play:

- **Product of Powers Rule**: When multiplying terms with the same base, add the exponents.- **Power of a Power Rule**: To raise a power to another power, multiply the exponents.- **Negative Exponent Rule**: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.
In our specific problem, we saw these rules in action when simplifying the expression \((x-1)^{\frac{1}{3} - (-\frac{2}{3})}\) to \((x-1)^{1}\). Understanding and correctly applying exponent rules is fundamental to simplifying expressions and making them more manageable.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations, and form the basis for much of algebra. Simplifying these expressions involves operations such as addition, subtraction, multiplication, and factoring. The goal is often to rewrite these expressions into a simpler or more useful form. In the problem provided, simplifying the expression involved factoring out a common element and then simplifying the resulting terms.

When working with algebraic expressions:

• Look for any patterns or common elements that can be simplified.
• Apply arithmetic operations and exponent rules accurately.
• Simplify terms step by step to avoid errors.

Simplification not only makes expressions easier to handle, but it also often reveals properties and solutions that might not be initially apparent.