Problem 33 - Elementary and Intermediate Algebra

Question

Find each product. See Example 2. $$ \left(0.4-9 m^{2}\right)\left(0.4+9 m^{2}\right) $$

Step-by-Step Solution

Answer

The product is \(0.16 - 81m^4\).

Step 1: Identify the Expression

The expression to be multiplied is \((0.4 - 9m^2)(0.4 + 9m^2)\). This is a multiplication of two binomials that are conjugates of each other.

Step 2: Apply the Difference of Squares Formula

Recognize that the expression is in the form \((a - b)(a + b)\), which can be simplified using the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\).

Step 3: Identify the Terms for a and b

In the expression \((0.4 - 9m^2)(0.4 + 9m^2)\), let \(a = 0.4\) and \(b = 9m^2\).

Step 4: Square the Terms

Square both terms: Calculate \(a^2 = (0.4)^2\) and \(b^2 = (9m^2)^2\). This gives us \(a^2 = 0.16\) and \(b^2 = 81m^4\).

Step 5: Substitute into the Formula

Substitute \(a^2\) and \(b^2\) into the difference of squares formula: \((0.4)^2 - (9m^2)^2 = 0.16 - 81m^4\).

Step 6: Simplify the Expression

After substituting, the expression \(0.16 - 81m^4\) is already simplified, as there are no like terms to combine.

Key Concepts

Difference of Squares

Algebra can sometimes seem daunting, but understanding key patterns can make things much easier. One such useful pattern is the "Difference of Squares." This is a special formula that allows us to simplify expressions where two conjugate binomials are involved. The classic formula is:

• \((a - b)(a + b) = a^2 - b^2\)

Here's what that means: whenever you see two binomials (terms with two parts) that are almost identical except for the sign between them (one positive, one negative), you can apply this formula. You multiply the first terms and then subtract the product of the second terms.
This formula is extremely handy because it saves time and reduces the complexity of calculations. For instance, instead of multiplying out every term, recognizing the pattern lets you find the product of two expressions much more efficiently. By using difference of squares, the multiplication simplifies almost immediately to a subtraction of squared terms. Whenever working algebraically, it’s useful to be on the lookout for such patterns as they frequently appear.

Binomials

Binomials are at the core of many algebraic expressions. A binomial is simply a polynomial that consists of exactly two terms. For example,

• \(0.4 - 9m^2\) is a binomial

Binomials are fundamental in algebra because they serve as the building blocks for more complex expressions. Operations involving binomials are common, including addition, subtraction, and multiplication. In our example exercise, the binomials also happen to be conjugates. Conjugate binomials are pairs like

• \((a - b)\) and \((a + b)\)

When multiplied, these binomials utilize the difference of squares formula.
Recognizing and understanding how to manipulate binomials using algebraic rules is crucial when dealing with more advanced polynomials and solving equations. As you practice with binomials, you'll become adept at spotting patterns and simplifying expressions efficiently.

Polynomial Multiplication

Polynomial multiplication underpins many algebraic processes, including the example exercise provided. When we talk about multiplying polynomials like binomials, we are essentially distributing each term across one another. This means, every term of the first polynomial multiplies with every term of the second polynomial. However, certain techniques, like recognizing the difference of squares, can drastically simplify the process.
Generally, multiplying polynomials involves using the distributive property, which may result in a large number of individual terms to simplify. But with expressions like

• \((0.4 - 9m^2)(0.4 + 9m^2)\)

where binomials are conjugates, recognizing this pattern allows us to use the difference of squares, neatly simplifying the multiplication into just two terms:

• \(0.16 - 81m^4\)

Such shortcuts are essential in algebra to manage complexity and keep calculations straightforward. By practicing polynomial multiplication, you get better at identifying these shortcuts which can be a huge timesaver, especially during tests.