Problem 71 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ x^{4}-4 x^{2} $$

Step-by-Step Solution

Answer

\(x^4 - 4x^2 = x^2(x + 2)(x - 2)\)

Step 1: Identify the Common Factor

First, observe that both terms in the polynomial \(x^4 - 4x^2\) share the common factor \(x^2\). Thus, we can factor out \(x^2\), which gives us:\[x^2(x^2 - 4)\]

Step 2: Recognize the Difference of Squares

The expression obtained from Step 1 is \(x^2(x^2 - 4)\). Notice that \(x^2 - 4\) is in the form of a difference of squares: \(a^2 - b^2\) where \(a = x\) and \(b = 2\). This can be factored further into \((a + b)(a - b)\).

Step 3: Factor the Difference of Squares

Apply the difference of squares formula learned in Step 2 to factor \(x^2 - 4\) as follows:\[x^2 - 4 = (x + 2)(x - 2)\]

Step 4: Write the Complete Factorization

Combine the results from the previous steps to express the original polynomial as a product of its factors. Keep the common factor \(x^2\) and substitute the factored form of \(x^2 - 4\) to get:\[x^2(x + 2)(x - 2)\]

Step 5: Verify the Factorization

Expand the factored expression to ensure it equals the original polynomial. Distribute \(x^2\) through the binomial \((x + 2)(x - 2)\), which gives:\[x^2(x + 2)(x - 2) = x^2(x^2 - 4) = x^4 - 4x^2\] The factorization is correct.

Key Concepts

Common Factor

When factoring polynomials, identifying and extracting the common factor is often the first and crucial step. A common factor is a term that is shared among the components of the polynomial. In our example, the polynomial given is \(x^4 - 4x^2\). Both terms, \(x^4\) and \(-4x^2\), share \(x^2\) as a common factor.

• Extract \(x^2\) from both terms.
• This reduces the expression to \(x^2(x^2 - 4)\).

Recognizing and factoring out common terms simplifies the polynomial and often makes additional factoring steps more straightforward.

Difference of Squares

The difference of squares is a special factorization form that occurs with expressions like \(a^2 - b^2\). The difference of squares formula is \((a+b)(a-b)\). In the expression \(x^2(x^2 - 4)\), the term \(x^2 - 4\) can be viewed as a difference of squares because:

• \(x^2 = (x)^2\)
• \(4 = (2)^2\)

Apply the formula:

• The expression \(x^2 - 4\) factors to \((x+2)(x-2)\).

This step gradually reveals a fully factored expression from what seemed a more complex polynomial.

Polynomials

Polynomials consist of terms that are comprised of variables raised to whole number powers, multiplied by coefficients. In our case, the polynomial is \(x^4 - 4x^2\). Some key properties include:

• Terms: Monomials added together, such as \(x^4\) and \(-4x^2\).
• Degree: Defined by the highest exponent of the variable, which here is 4.
• Constant: An optional component with a degree of 0, although our example has none.

Understanding these basics helps in recognizing patterns that allow easier manipulation and simplification when factoring.

Algebra

Algebra provides the framework for manipulating symbols and solving equations, which includes factoring polynomials. Important concepts in algebraic factorization involve

• Variable manipulation,
• Recognizing patterns like the common factor and the difference of squares,
• And verifying solutions through expansion and substitution.

For instance, once the expression \(x^4 - 4x^2\) is factored to \(x^2(x+2)(x-2)\), a simple algebraic check by re-expanding proves that it indeed matches the original polynomial.

Mathematics

Mathematics, the broad discipline encompassing algebra, is a way to solve complex problems through logical reasoning and techniques. Factoring is one of many techniques used for simplifying expressions and equations. It highlights:

• The power of recognizing simple patterns,
• The systematic approach to solving polynomial equations,
• And the importance of step-by-step logical reasoning to validate solutions.

Overall, recognizing opportunities for factoring transforms mathematical expressions into simpler, more manageable forms, facilitating problem-solving across various fields of mathematics.