Problem 50 - Algebra A Combined Function

Question

Factor each completely. $$ x^{2}+20 x+100-x^{4} $$

Step-by-Step Solution

Answer

Factor: \(-(x^2 - 10)^2\).

Step 1: Rearrange the polynomial

First, recognize the polynomial given: \(-x^4 + x^2 + 20x + 100\). Notice it can be rearranged for easier manipulation. Rewrite it as: \(-x^4 + x^2 + 20x + 100\).

Step 2: Factor out common terms

Look for any common terms in the polynomial. Notice that no common factor exists across all terms. However, the expression can initially be influenced by recognizing it contains squared terms and perhaps can be rewritten as squares: \((x^2)^2\).

Step 3: Recognize it as a difference of squares

Notice the terms \(x^4\) and \(100\) can form a difference of squares. Reorganize the expression as \(-(x^4 - x^2 - 20x - 100)\). Now see if further factoring can be done under the current rearrangement.

Step 4: Apply difference of squares

The expression inside the parenthesis can be viewed as the difference of squares if rewritten as \((x^2 - 10)^2\). Thus the original equation becomes \(-(x^2-10)^2\). This confirms a difference where the square of \(x^2\) and 10 are distinct, proving this within \(-(x^2-10)^2\).

Step 5: Verify factorization

Multiply back the factors to verify the results match the original polynomial \(-x^4 + x^2 + 20x + 100\). This verifies the factorization as \(-(x^2 - 10)^2\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a unique factorization technique that simplifies expressions where two perfect squares are subtracted. It is defined as \[a^2 - b^2 = (a + b)(a - b)\]. This formula indicates that the difference between two squares can always be split into two binomial factors.
In the context of our polynomial, we observe terms that can be interpreted as squares: \(x^4\) can be seen as \((x^2)^2\) and 100 as \((10)^2\). Understanding these as perfect squares allows us to rearrange and factor the expression as a difference of squares. Recognizing these squares helps restructure the given polynomial into a form that's easier to manipulate, leading to the solution of \(-(x^2 - 10)^2\) for the original expression.
Observe how applying this concept effectively reduces complex expressions to simpler, yet equivalent forms.

Factoring Techniques

Factoring techniques are powerful mathematical tools used to simplify expressions or equations by breaking them down into more manageable components.
There are several techniques available:

• Extracting common factors: Where all terms in the expression share a common factor, which can be factored out.
• Grouping: This involves rearranging and grouping terms so that they become more factorable.
• Special products: Recognize and utilize formulas for squares, cubes, and other polynomial structures, including the difference of squares.

In this exercise, after identifying no common factor directly across all terms, we rely on recognizing the structure of the polynomial itself. By noticing \(x^4 - 100\) as a difference of squares, we applied a technique suited to this specific form.
Choosing the right factoring technique often depends on recognizing patterns and understanding the underlying structure of the polynomial.

Rearranging Polynomials

Rearranging polynomials involves changing the order of terms in the expression to make it easier to work with or to recognize certain patterns.
For successful factoring and simplification, it often requires one to recognize potentially hidden structures. In the given exercise, after rewriting the original polynomial as \(-x^4 + x^2 + 20x + 100\), we analyzed terms to identify underlying patterns or special cases.
The goal of such rearrangement is hardly to change the value of a polynomial but rather to represent it in ways optimal for techniques like factoring.

• Identify structures: Look for familiar patterns (like squares or cubes).
• Simplification: Rewrite expressions to highlight specific terms or arrangements.

Thus, rearranging can often be a pivotal first step in solving polynomial equations as seen when we rearranged terms seeking to set up the process of identifying a difference of squares.