Problem 86 - Algebra and Trigonometry

Question

85–90 ? Factor the expression completely. (This type of expression arises in calculus when using the “product rule.”) $$ 5\left(x^{2}+4\right)^{4}(2 x)(x-2)^{4}+\left(x^{2}+4\right)^{5}(4)(x-2)^{3} $$

Step-by-Step Solution

Answer

The fully factored expression is \((x^2 + 4)^4 (x - 2)^3 (14x^2 - 20x + 16)\).

Step 1: Identify Common Terms

Look for common factors in both terms of the expression. Both terms contain \( (x^2 + 4)^4 \) and \( (x - 2)^3 \) as part of their factors.

Step 2: Factor Out Common Terms

Factor out the common terms \( (x^2 + 4)^4 (x - 2)^3 \) from the entire expression.\[(x^2 + 4)^4 (x - 2)^3 \, [5 \cdot (2x) \cdot (x - 2) + 4(x^2 + 4)]\]

Step 3: Simplify the Remaining Part

Simplify the expression inside the brackets:- First term: \(5 \cdot 2x \cdot (x - 2) = 10x(x - 2)= 10x^2 - 20x\)- Second term: \(4(x^2 + 4) = 4x^2 + 16\)So the simplified expression is:\[ 10x^2 - 20x + 4x^2 + 16 \]

Step 4: Combine and Simplify Like Terms

Combine the like terms:\[(10x^2 + 4x^2) + (-20x) + 16 = 14x^2 - 20x + 16\]

Step 5: Distribute

Write the fully factored expression:\[(x^2 + 4)^4 (x - 2)^3 (14x^2 - 20x + 16)\]No further factorization is immediately apparent, but you can check for any simplification steps.

Key Concepts

Product Rule

When dealing with calculus, the product rule is a technique used to find the derivative of a product of two functions. Here in algebra, the term "product rule" often refers to the way terms in an expression are multiplied or factored together. For example, when an expression like \( 5\left(x^{2}+4\right)^{4}(2 x)(x-2)^{4} \) contains multiple terms multiplied by each other, it showcases the product rule concept.

• Recognize that you are multiplying several expressions or terms together.
• Each part is considered separately before combining in the final equation.

Understanding this concept helps in breaking down complex algebraic expressions, making them easier to manipulate and simplify.

Common Factors

One of the most crucial steps in simplifying an algebraic expression is identifying common factors. A common factor is a term that appears in every part of an expression or equation. In our example, both components of the expression have \((x^2 + 4)^4 \) and \((x - 2)^3 \).

• Identifying common factors can significantly reduce the complexity of an expression.
• Always look at each term closely to find these shared elements.

Once found, factoring out these common elements can make combining remaining terms more straightforward, as it essentially "cleans up" the equation by removing these repeated sections, thus simplifying the expression.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form, often by combining like terms. Simplifying the polynomial inside the brackets of our expression means dealing with terms like \(10x^2 - 20x + 4x^2 + 16\).

• Identify terms with the same variable power ("like terms").
• Combine these like terms to simplify your polynomial.

In this example, combining like terms transforms \(10x^2 + 4x^2\) into \(14x^2\), and leaves the entire expression clearer and easier to understand. Simplifying further may sometimes uncover new factoring opportunities, but in this case, additional factorization wasn’t immediately apparent.

Distribute

The concept of distribution in algebra involves spreading a common factor over the terms inside an expression. After factoring out common terms, such as in \((x^2 + 4)^4 (x - 2)^3 (14x^2 - 20x + 16)\), remember to "distribute" these terms correctly.

• Ensure each term inside the parenthesis is multiplied by the factors outside when you reassemble the equation.
• This step is crucial in writing the expression in its fully factored form, ensuring nothing is left out.

Distribution helps in setting up the expression for further operations or evaluations, making it vital for ensuring accuracy in algebraic manipulations.