Problem 61 - Beginning and Intermediate Algebra

Question

Factor completely. $$144 m^{2}-n^{4}$$

Step-by-Step Solution

Answer

The completely factored expression is \((12m - n^2)(12m + n^2)\).

Step 1: Identify the difference of squares

The given expression is \(144m^2 - n^4\). We can rewrite it as \((12m)^2 - (n^2)^2\). Now, we can recognize that it is a difference of squares, with \(a = 12m\) and \(b = n^2\).

Step 2: Apply the difference of squares formula

Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we can substitute 'a' and 'b' with the values we found in Step 1: \((12m - n^2)(12m + n^2)\) Now, we have factored the expression as a difference of squares.

Step 3: Check if further factoring is possible

We need to check if there's any possible further factoring for the terms inside the parentheses. In this case, there isn't any, since we don't have any common factors between the coefficients and variables inside the parentheses. So, the expression is factored completely as: \((12m - n^2)(12m + n^2)\)

Key Concepts

Difference of Squares

The difference of squares is an important algebraic identity used in factoring. This technique is applied when you have an expression that fits the form \(a^2 - b^2\). It's called "difference of squares" because you're subtracting one perfect square from another. This identity is useful because it allows us to express the difference of two squares as a product of two binomials.

• Using the formula \(a^2 - b^2 = (a - b)(a + b)\), you can break down complex expressions into simpler products.
• Once the terms are in square form, you can easily transform them into factored form.

In our exercise, \((12m)^2 - (n^2)^2\) represents \((a^2 - b^2)\), where \(a = 12m\) and \(b = n^2\). By recognizing this, we can quickly apply the formula to factor it into \((12m - n^2)(12m + n^2)\). This simplification is often a major step in solving polynomial equations.

Polynomial Expressions

Polynomial expressions are a foundational concept in algebra and mathematics as a whole. They are mathematical expressions involving variables, coefficients, and exponents, combined using operations such as addition, subtraction, and multiplication.

• A polynomial expression can consist of multiple terms, which are parts of the expression separated by plus or minus signs.
• Each term in a polynomial includes a variable raised to an integer power and multiplied by a coefficient.

In the given exercise, \(144m^2 - n^4\) is a polynomial expression. It has two terms: \(144m^2\) and \(-n^4\). Recognizing the structure of these terms is crucial for employing factoring techniques. Factoring polynomial expressions is a valuable skill, especially when solving equations or simplifying mathematical problems.

Factoring Techniques

Factoring is an essential technique in algebra for breaking down expressions into simpler, more manageable pieces. Different factoring techniques apply depending on the structure of the polynomial.

• The **difference of squares** is one technique, especially when dealing with binomials that fit the \(a^2 - b^2\) form.
• Other techniques include factoring out the greatest common factor (GCF), factoring by grouping, and factoring trinomials.

In the step-by-step solution provided for the expression \(144 m^{2}-n^{4}\), the difference of squares technique was used because the expression could be rewritten as \((12m)^2 - (n^2)^2\). It's crucial to first identify the appropriate technique based on the form of the expression. Once identified, applying the factoring method can significantly simplify the problem, leading to easier manipulation of polynomial expressions and solutions to equations. By mastering different factoring techniques, you enhance your problem-solving toolkit in algebra.