Problem 3 - Elementary Algebra

Question

If possible, factor the following binomials completely. $$ 49 a^{4}-b^{2} c^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the given binomial expression is $$(7a^2 + b^2c^2)(7a^2 - b^2c^2)$$.

Step 1: Identify the difference of squares pattern

The given expression can be rewritten as $$(7a^2)^2 - (b^2c^2)$$ which indicates that it is in the form of a difference of squares. The difference of squares formula states that for any expressions p and q, $$p^2 - q^2 = (p + q)(p - q)$$. In our case, \(p = 7a^2\) and \(q = b^2c^2\).

Step 2: Apply the difference of squares formula

Using the difference of squares formula, the factored form of the given expression will be $$(7a^2 + b^2c^2)(7a^2 - b^2c^2)$$ So the factored form of the given binomial is $$(7a^2 + b^2c^2)(7a^2 - b^2c^2)$$.

Key Concepts

Binomial Factoring

Factoring is a foundational concept in algebra that involves breaking down an algebraic expression into simpler factors that, when multiplied together, give back the original expression. A binomial is a polynomial with exactly two terms, and the process of factoring binomials often reveals interesting patterns that can be applied to solve various algebra problems.

Here's how you can approach binomial factoring: first, look for a common factor in both terms. If one exists, factor it out. Next, check if the binomial fits any special factoring formulas, such as the difference of squares, the sum and difference of cubes, or the difference of squares. Recognizing these patterns is crucial for efficient factoring. For students, understanding these fundamental patterns helps simplify more complex factoring challenges in later studies of algebra and calculus. When factoring, ensure that the multiplication of the factors indeed gives you the original binomial to confirm the factoring is correct.

Difference of Squares Pattern

The difference of squares is a specific type of binomial that can be easily factored if you recognize its structure. This pattern occurs when a binomial is composed of two terms: one is a square of an expression, and the other is the negative square of another expression. The general form of a difference of squares is \(a^2 - b^2\).

To factor a difference of squares, you can use the formula:\[ a^2 - b^2 = (a + b)(a - b) \]. In practical terms, when you identify a binomial in this form, you simply find \(a\) and \(b\) and plug them into this factoring formula. This method is powerful because it applies to any expressions where each term is a perfect square and they are subtracted from each other. Using the difference of squares to factor binomials not only makes the problems easier to tackle but also aids in understanding the underlying structure of algebraic equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \), \( y \) or \( a^2 \)), and operators (like add, subtract, multiply, and divide). These components combine to represent a specific mathematical quantity. An important aspect of algebraic expressions is that they don't contain an equals sign, distinguishing them from equations.

Algebraic expressions are the building blocks of algebra and are often manipulated through various operations. Understanding how to work with these expressions enables you to solve many different mathematical problems. When dealing with algebraic expressions, remember to be attentive to the order of operations and to manipulate all parts of the expression according to the algebraic rules that have been established like combining like terms and using distribution laws. This understanding is key for successfully mastering more complex areas of mathematics.