Problem 42 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$-4 x^{2}+4$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(-4 x^{2}+4\) is -4\((x-1)(x+1)\)

Step 1: Identify the common factors

Taking a look at the given polynomial \(-4 x^{2}+4\), it can be observed that 4 is a common factor for both terms.

Step 2: Factoring out the common factor

Factoring out the common term, the equation becomes -4(\(x^{2}-1\))

Step 3: Recognize the Difference of Squares

The equation inside the brackets \(x^{2}-1\) is recognized as difference of squares. The difference of squares is a special case in factoring and it follows this pattern: \[a^2 - b^2 = (a+b)(a-b)\]

Step 4: Apply the formula

Applying the difference of squares formula to \(x^{2}-1\), it can be factored to \((x-1)(x+1)\). Hence \(-4(x^{2}-1)\) can be rewritten as -4\((x-1)(x+1)\)

Key Concepts

Common Factor

When approaching any polynomial expression, the simplest starting point is to identify if there is a common factor shared across all its terms. It means looking for a number or variable that can be evenly divided into each term. In our example, from the polynomial \(-4x^2 + 4\), we identify that the number 4 is common to both \(-4x^2\) and \(+4\). Extracting the common factor can often simplify the polynomial and make subsequent factoring steps more manageable.

• Why Identify a Common Factor: It reduces the complexity of the polynomial, making further steps easier to execute.
• How to Identify: Check what maximum number or variable(s) can divide each term fully without leaving a remainder.
• Example: Starting with \(-4x^2 + 4\), factor the 4, resulting in \(-4(x^2 - 1)\).

Difference of Squares

The difference of squares is a specific type of polynomial that takes the form \(a^2 - b^2\). It is characterized by two terms being squared and separated by a subtraction sign. This pattern provides a handy shortcut for factoring because it can always be written as the product of two binomials.
In our example, once we factor out the common factor and get the expression \(x^2 - 1\), we notice it fits the difference of squares pattern:\[(x^2 - 1) = (x^2 - 1^2)\]

• Recognizing the Pattern: Look for two terms that are perfect squares separated by a subtraction sign.
• The Formula: The difference of squares can be expressed as \((a+b)(a-b)\).
• Applying the Formula: For \(x^2 - 1\), this becomes \((x+1)(x-1)\).

This method provides a quick way to simplify and solve expressions that might otherwise seem complex.

Factoring Techniques

Factoring polynomials like \(-4x^2 + 4\) involves using different techniques to break down a polynomial into simpler, multipliable terms. These techniques, such as identifying a common factor and recognizing special patterns, are invaluable tools for simplifying expressions and solving algebraic equations more effectively.

• Common Factor: Always start by searching for common factors. It simplifies the polynomial through division, making it more approachable for further factoring.
• Special Patterns: Patterns like the difference of squares (\(a^2 - b^2\) pattern) can help break down complex expressions quickly.
• Factoring Step-by-Step: Start with identifying and factoring out any common factors. Once simplified, look for patterns like the difference of squares.

Every time you simplify a polynomial, you'll be employing these techniques either individually or in combination with others to reach a fully factorized answer. With practice, recognizing and applying these steps becomes quicker and more intuitive, easing the path to solving complex polynomials.