Problem 50 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 128-32 x^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the binomial expression $$128 - 32x^2$$ is $$32(2 + x)(2 - x)$$.

Step 1: Identify common factors

First, we need to identify if there are any common factors between the two terms in the binomial expression. In this case, both terms share a common factor of 32.

Step 2: Factor out the common factor

Now, factor out the common factor (32) from each term in the binomial expression: $$ 32(4 - x^2) $$

Step 3: Identify the difference of squares

In the factored expression, notice that the remaining binomial \((4 - x^2)\) is a difference of squares. A difference of squares can be factored further using the following identity: $$ a^2 - b^2 = (a + b)(a - b) $$

Step 4: Apply the difference of squares identity

Applying the difference of squares identity to the remaining binomial expression, we have: $$ 4 - x^2 = (2 + x)(2 - x) $$

Step 5: Combine the factored expressions

Finally, combine the factored expressions from Step 2 and Step 4 to find the fully factored form of the given binomial: $$ 128 - 32x^2 = 32(4 - x^2) = 32(2 + x)(2 - x) $$ So, the factored form of the given binomial expression is: $$ 128 - 32x^2 = 32(2 + x)(2 - x) $$

Key Concepts

Difference of Squares

The difference of squares is a special algebraic expression that can be identified by its unique form. It consists of two squared terms with a subtraction between them. Mathematically, it is expressed as \(a^2 - b^2\). The beauty of this expression is its simplicity when factoring. You can factor it using the identity:

• \(a^2 - b^2 = (a + b)(a - b)\)

In our given problem, the inner expression \(4 - x^2\) is a difference of squares. Here, \(4\) can be seen as \(2^2\) and \(x^2\) remains as \(x^2\). Applying this identity results in \((2 + x)(2 - x)\). Recognizing these patterns quickly helps in solving algebra problems efficiently.

Common Factor

Identifying a common factor is often the first step in factoring polynomials. A common factor is a number or expression that divides each term in the polynomial without leaving a remainder. In our exercise, both terms of the binomial \(128\) and \(32x^2\) share a common factor of \(32\), as both are divisible by \(32\).

• Checking for common factors simplifies expressions.
• It reduces the polynomial to a simpler form, making further factoring easier.

In this problem, dividing each term by \(32\) gives us \(32(4 - x^2)\), greatly simplifying the task.

Factoring Technique

Factoring is an essential technique in algebra that involves writing a polynomial as a product of its simplest parts, or factors. Different types of expressions require different approaches to factoring.

• Look for a common factor first—this simplifies the entire expression.
• Identify special patterns like the difference of squares to apply specific identities.

The provided example uses both these techniques seamlessly. It initially factors out a common number (\(32\)), then recognizes and applies the difference of squares identity to \(4 - x^2\), resulting in a fully factored expression \(32(2 + x)(2 - x)\). Mastering these techniques is crucial for tackling more complex algebraic problems.

Binomial Expression

A binomial is an algebraic expression containing exactly two terms, which can involve numbers, variables, or a combination of both. In the given expression, \(128 - 32x^2\) is a binomial because it consists of two distinct terms: \(128\) and \(-32x^2\).

• Recognizing binomials is a fundamental skill in algebra.
• Different factoring methods are applied based on the type of expression.

Understanding what makes up a binomial and differentiating them from other algebraic expressions is the first step in effectively applying factoring techniques.