Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the binomials. \((11 q-10)(11 q+10)\)

Step-by-Step Solution

Answer

The product of the binomials is \(121q^2 - 100\).

Step 1: Understanding the Problem

We need to multiply the binomials \((11q - 10)\) and \((11q + 10)\). This is a classic example of multiplying the sum and difference of two terms, which can be solved using a special product formula.

Step 2: Recognizing the Formula

The product of the sum and difference of the same two terms follows the formula: \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 11q\) and \(b = 10\).

Step 3: Applying the Formula

Substitute \(a = 11q\) and \(b = 10\) into the formula: \((11q)^2 - 10^2\).

Step 4: Calculating \((11q)^2\)

Calculate \((11q)^2\) which equals to \(121q^2\). This is obtained by multiplying \(11q\) by itself.

Step 5: Calculating \(10^2\)

Calculate \(10^2\) which equals to \(100\). This is simply \(10\) multiplied by \(10\).

Step 6: Finalizing the Expression

Substitute back into the equation: \(121q^2 - 100\). This simplified expression represents the product of the two binomials.

Key Concepts

Special Product Formula

When dealing with binomials like \((11q - 10)(11q + 10)\), we can make use of a powerful mathematical shortcut known as the special product formula. This formula focuses on the **product of the sum and difference of two terms**. Instead of expanding the binomials directly, you recognize that there's a specific structure involved,

• The formula is: \((a-b)(a+b) = a^2 - b^2\).
• This is called the **difference of squares** identity because multiplying these specific types of binomials results in a difference between the square of two terms.

Using this method simplifies the process and reduces the chance of errors commonly made when using the distributive property or the FOIL method. Here, if \(a = 11q\) and \(b = 10\),then the expression becomes \((11q)^2 - 10^2\), allowing for a straightforward computation.

Sum and Difference of Terms

The sum and difference of terms occur when you have two binomials, where one is the addition of two terms and the other is the subtraction of the same two terms. This forms the basis for using the special product formula. When you encounter a binomial of the form \((a+b)(a-b)\), remember that it is just the **sum and difference of the same terms**.

• In essence, they "cancel" each other out in terms of cross products, which is why you're left with a subtraction (or difference) in the final calculation.
• It's crucial to note that the middle terms cancel out, providing a cleaner and simplified solution.

Imagine the middle terms in these kinds of expressions cancel each other out, which makes this particular type of multiplication easier to handle.

Simplifying Expressions

Simplifying expressions involves reducing them to their most compact form,making them easier to handle and understand. In the context of multiplying binomials like \((11q - 10)(11q + 10)\),simplification means using the special product formula:

• After applying the formula, you arrive at \(a^2 - b^2\), which in cases like this, means replacing these with their computed squares.
• Calculate \((11q)^2 = 121q^2\) and \(10^2 = 100\).
• The last step is to write out the simplified expression: \(121q^2 - 100\). This indicates the final result of multiplying the binomials, which is the essence of simplifying expressions.

Simplification helps in seeing the beauty and symmetry in mathematics by ensuring that no further operations are needed, thus reflecting the exact and complete *product*.