Problem 101 - College Algebra

Question

Explain how to square a binomial difference. Give an example with your explanation.

Step-by-Step Solution

Answer

To square a binomial difference, you apply the FOIL method for multiplication. The result is \(a^2 - 2ab + b^2\). For example, with \(a = 3\) and \(b = 2\), \((3 - 2)^2 = (3 - 2)(3 - 2) = 1\).

Step 1: Explain the FOIL Method

The FOIL method stands for First, Outside, Inside, Last. It's a process used for multiplying binomials. In the case of the binomial difference, which is \((a - b)\), when it is squared it becomes \((a - b)(a - b)\). The FOIL method instructs you to multiply the first terms in each binomial together, then the outsides, then the insides, then the last terms.

Step 2: Apply the FOIL Method

Using the FOIL method with \((a - b)^2 = (a - b)(a - b)\), the first terms are both \(a\), the outside terms are \(a\) and \(-b\), the inside terms are \(-b\) and \(a\), and the last terms are both \(-b\). Hence you get: \(a * a = a^2\), \(a * -b = -ab\), \(-b * a = -ab\), and \(-b * -b = b^2\).

Step 3: Combine Like Terms

The terms \(-ab\) and \(-ab\) are alike and can be combined. So the final expression after multiplying out the binomial difference and combining like terms is: \(a^2 - 2ab + b^2\).

Step 4: Example with specific values

To illustrate this, let's use the specific values \(a = 3\) and \(b = 2\) to form the binomial difference and square it. The process is carried out the same way as the general case. So, \((3 - 2)^2 = (3 - 2)(3 - 2) = 3^2 - 2*3*2 + 2^2 = 9 - 12 + 4 = 1\).