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Chapter 0: Problem 101

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(-b * -b = b^2\).
03

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: \(a^2 - 2ab + b^2\).
04

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, \((3 - 2)^2 = (3 - 2)(3 - 2) = 3^2 - 2*3*2 + 2^2 = 9 - 12 + 4 = 1\).

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