Question

Challenge Problems.$$\left(2.83 m^{2}-2.12 n^{2}\right)\left(2.83 m^{2}+2.12 n^{2}\right)$$

Short answer

\[ 8.0089 m^4 - 4.4944 n^4 \]

Step-by-step solution

Identify the Pattern

Recognize that the given expression follows the pattern of a difference of squares, which is \[a^2 - b^2 = (a - b)(a + b)\], where in this case, \(a = 2.83 m^2\) and \(b = 2.12 n^2\).

Apply the Difference of Squares Formula

Substitute the values into the difference of squares formula to obtain \[ (2.83 m^2)^2 - (2.12 n^2)^2 = (2.83 m^2 - 2.12 n^2)(2.83 m^2 + 2.12 n^2) \].

Simplify the Expression

Simplify the expression by multiplying the terms which results in \[ (2.83 m^2)^2 - (2.12 n^2)^2 = 8.0089 m^4 - 4.4944 n^4 \].

Key concepts

Polynomial Multiplication

Polynomial multiplication is a key concept in algebra that involves the multiplying of two or more polynomials to produce a new polynomial. It's like combining several strings of numbers and variables, each raised to a power, into one. When faced with an expression like the one from our exercise (2.83 m^{2}-2.12 n^{2})(2.83 m^{2}+2.12 n^{2}), it's important to identify each term of the polynomial and apply the appropriate multiplication rules.

For instance, when multiplying two binomials, you can use the FOIL method, which stands for First, Outside, Inside, Last, referring to the terms of each binomial. However, when a pattern such as the difference of squares is recognized, you can streamline the process. By applying this specific algebraic identity, you effectively reduce the complexity of the multiplication, as demonstrated in the textbook solution.

Understanding how to multiply polynomials will not only help in solving algebraic expressions but also set a solid foundation for more advanced mathematical concepts, like calculus, where polynomial functions play a significant role.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. They are useful shortcuts to simplify expressions and solve equations efficiently. One of the most fundamental identities is the difference of squares, which is expressed as a^2 - b^2 = (a - b)(a + b). This identity illustrates that the difference between two squared terms can be factored into the product of two binomials.

In the given problem, identifying that (2.83 m^{2})^{2} - (2.12 n^{2})^{2} fits the difference of squares identity simplifies the process considerably and avoids the lengthy standard polynomial multiplication. Familiarity with algebraic identities is not just a matter of memorization, but understanding how they are derived can be a powerful tool in a student's mathematical toolset, aiding in the simplification of complex algebraic challenges.

Simplifying Expressions

Simplifying expressions is a frequent task in algebra that involves reducing an expression to its simplest form while keeping it mathematically equivalent. This often includes combining like terms, factoring, and utilizing algebraic identities to minimize the number of terms and complexity of expressions. In the context of our textbook problem, simplifying the given expression involves recognizing the structure of the difference of squares and applying that knowledge to find a more straightforward solution.

After applying the difference of squares formula, we reach the new expression 8.0089 m^4 - 4.4944 n^4, which is much simpler than the original expanded product would have been. It's crucial for students to grasp these techniques, as simplifying expressions not only makes mathematical problems more manageable but also helps in better understanding and solving various real-world problems.