Problem 106 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$b x^{2}-4 b+a x^{2}-4 a$$

Step-by-Step Solution

Answer

The given polynomial \(b x^{2}-4 b+a x^{2}-4 a\) can be factored completely as \((a+b)(x^{2}-4)\)

Step 1: Rearrange the Polynomial

Rearrange the terms in the expression to group like terms together: \(a x^{2} + b x^{2} - 4a - 4b\)

Step 2: Factor out the Common Factors

Factor out common factors from each group of two terms. From the first two terms \(a x^{2} + b x^{2}\), we can factor out \(x^{2}\) to get \((a+b) x^{2}\). From the last two terms \(-4a - 4b\) we can factor out \(-4\) to get \(-4(a+b)\). So the expression becomes: \((a+b) x^{2} - 4(a+b)\)

Step 3: Final Factorization

Again observe common factor \((a+b)\) in both the terms in the expression. Factor it out and the expression becomes: \((a+b)(x^{2}-4)\)

Step 4: Check by Multiplying

Perform the multiplication operation to ensure the factoring was done correctly. \((a+b)(x^{2}-4)\) = \(a x^{2} + b x^{2} - 4a - 4b\). This matches with the original arranged polynomial, hence the factoring was done correctly.

Key Concepts

Polynomial Factorization

Polynomial factorization is a critical process in algebra where a polynomial is expressed as the product of its factors. It's like breaking down a complex structure into simpler pieces that are easier to understand and manage. For example, take the polynomial equation \(b x^{2}-4 b+a x^{2}-4 a\). The goal is to rewrite it in a manner that resembles \( (something) \times (something else) \), making it clear how the terms relate to each other. Factorization helps in solving equations, simplifying expressions, and finding roots of equations.

One useful strategy for factorization is looking for common factors across terms, which can be then factored out. Systematically combining like terms and rearranging the polynomial can reveal more factorization opportunities. As seen in the provided solution steps, rearranging and combining like terms simplified the polynomial, allowing the common factors from groups of terms to be extracted. Always remember to verify your factorization by multiplying the factors out to ensure they produce the original polynomial.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and division. In the case of \(b x^{2}-4 b+a x^{2}-4 a\), it represents an algebraic expression involving two variables, \(a\) and \(b\), and the square of another variable, \(x\). Understanding the structure of algebraic expressions is essential to manipulate and simplify them through processes such as factoring.

The complexity of algebraic expressions can often be reduced by combining like terms and finding common factors, which leads to a more elegant and simple form of the expression and can be particularly useful when solving equations. The well-arranged expression not only makes the calculations more tangible but also reveals the inherent properties of the mathematical relationships involved.

Like Terms

In algebra, like terms refer to terms that contain the same variables raised to the same power. These terms can be combined – added or subtracted – due to their similar structure. For instance, in the expression \(a x^{2}\) and \(b x^{2}\), both terms are considered like terms because the variable \(x\) is squared in each. When you're trying to factor polynomials, grouping like terms simplifies the process by reducing clutter and highlighting commonality.

In our exercise, recognizing that \(a x^{2}\) and \(b x^{2}\) are like terms is the first step toward simplification. This insight is a small, yet crucial, aspect of polynomial factorization as it sets the stage for factoring by grouping. Ensuring that you are comfortable with identifying and combining like terms will greatly aid in your study of algebra.

Common Factors

Common factors are numbers or expressions that divide exactly into each term of a given algebraic expression. Identifying common factors is an important skill in the factorization process, as it enables you to simplify expressions by removing redundancy. In the polynomial \(a x^{2} + b x^{2} - 4a - 4b\), the term \(x^{2}\) is a common factor in the first two terms, while \(4\) (or specifically, \( -4\)) is a common factor in the last two terms. By factoring out these common elements, the expression is drastically simplified.

Finding common factors can sometimes seem challenging, but by systematically pairing terms and looking for the greatest common factor in each pair, the process becomes more manageable. Once the common factors are factored out and you've achieved a simpler form, always cross-check by multiplying the factors to see if you get back to the original expression. This verification step is crucial in ensuring the accuracy of your factorization.