Problem 29 - Algebra 2 and Trigonometry

Question

In \(27-39,\) factor each polynomial completely. $$ b^{3}-4 b $$

Step-by-Step Solution

Answer

The polynomial is factored as \( b(b+2)(b-2) \).

Step 1: Identify Common Factors

Look at the terms of the polynomial \( b^3 - 4b \). Identify any common factor in both terms. Both terms contain the factor \( b \).

Step 2: Factor Out the Common Factor

Factor \( b \) out of both terms: \( b(b^2 - 4) \). This simplifies the expression by removing the common factor from each term.

Step 3: Recognize the Difference of Squares

Understand that \( b^2 - 4 \) is a difference of squares, since \( 4 \) is \( 2^2 \) and can be rewritten as \( (b + 2)(b - 2) \). The original expression now looks like this: \( b(b+2)(b-2) \).

Step 4: Verify Factorization

Multiply the factors back together to ensure the original polynomial is achieved: \( b(b + 2)(b - 2) = b(b^2 - 4) = b^3 - 4b \). This confirms the factorization is correct.

Key Concepts

Difference of Squares

When factoring polynomials, the difference of squares is a pattern that can simplify expressions dramatically. The difference of squares refers to an expression in the form of \( a^2 - b^2 \). In this case, it's the subtraction of two perfect squares. A perfect square is a number or expression that can be written as something squared, like \( 4 \) which is \( 2^2 \). This form allows for a straightforward factorization into \((a + b)(a - b)\).

In the given polynomial \( b^3 - 4b \), once the common factor \( b \) is factored out, we identify \( b^2 - 4 \) as a difference of squares, since \( b^2 \) is already a square \( (b)^2 \) and \( 4 \) is \( 2^2 \). Thus, it can be rewritten as \( (b + 2)(b - 2) \), completing the factorization of the polynomial.

Common Factor

A common factor is a term that divides each term in a polynomial without leaving a remainder. Finding a common factor is often the first step in factoring polynomials. It's like pulling apart the pieces of a puzzle to see a clearer picture.

In the polynomial \( b^3 - 4b \), we note that each term contains the factor \( b \). By dividing \( b^3 \) by \( b \), we obtain \( b^2 \) and dividing \( -4b \) by \( b \) gives us \( -4 \). This makes \( b \) the common factor. Pulling out this \( b \) simplifies the polynomial to \( b(b^2 - 4) \). Recognizing common factors simplifies the structure and is a key skill in algebra.

Polynomial Expressions

Polynomial expressions represent sums of terms that each consist of a variable raised to a non-negative integer power. These expressions can be categorized by the number of terms they have. Each type has its own unique properties and faceting methods.

• Monomial: A single term with a variable, like \( 5x^2 \).
• Binomial: Two terms, e.g., \( x^2 - 1 \).
• Trinomial: Three terms, such as \( x^2 - 4x + 4 \).

Recognizing these types will aid in choosing the appropriate method to simplify. For our specific polynomial \( b^3 - 4b \), initially, it features two terms which can be seen as a binomial. The process of factoring involves recognizing the structure, identifying common factors, and looking for patterns like the difference of squares. This structured approach leads to properly breaking down and simplifying polynomial expressions.