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Chapter 4: Problem 123

Describe the steps involved in finding the product of a binomial and its conjugate.

Short Answer

Expert verified
The product of a binomial and its conjugate is found by multiplying the binomial by its conjugate and simplifying the result which leads to the difference of squares: \(a^2 - b^2\).

Step by step solution

01

Understanding the Binomial and its Conjugate

Firstly, identify the binomial and its conjugate. If your binomial is \(a + b\) , the conjugate of this binomial will be \(a - b\). Similarly, if your binomial is \(a - b\), the conjugate will be \(a + b\). The only difference between a binomial and its conjugate is the sign between the two terms.
02

Multiplying the Binomial to its Conjugate

The multiplication of a binomial and its conjugate falls under the FOIL method (First, Outer, Inner, Last). While multiplying, multiply each pair: \((a + b)\) and \((a - b)\). This will result in \(a^2 - ab + ab - b^2\).
03

Simplifying the Product

The result of step 2 will lead to cancellation of middle terms due to the negative and positive terms being the same. That means \(a^2 - ab + ab - b^2\) will simplify to \(a^2 - b^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binomials
A binomial is a type of algebraic expression that contains exactly two terms, which are usually separated by a plus or minus sign. These expressions can look like \(a + b\) or \(a - b\). Binomials are the building blocks of many algebraic operations. Understanding them is crucial for further operations such as addition, subtraction, and importantly, multiplication.

Here are key things to remember about binomials:
  • Binomials always have two distinct terms.
  • Each term in a binomial may themselves consist of variables, coefficients, or both.
  • Binomials are used in forming polynomials, which are expressions with more than one term.
When dealing with binomials, often you鈥檒l encounter their conjugates. Conjugates can help you simplify expressions, especially when it comes to multiplication.
Conjugates in Algebra
Conjugates are special because they allow for simplifying products in algebra, particularly when dealing with binomials. For a binomial \(a + b\), its conjugate is \(a - b\). Conversely, the conjugate of \(a - b\) is \(a + b\). The change in sign between the two terms of a binomial is what defines its conjugate.

Conjugates have several properties useful in algebra:
  • When a binomial is multiplied by its conjugate, the result is a difference of squares, namely \(a^2 - b^2\). This eliminates mixed terms, simplifying the expression.
  • Conjugates are often used in rationalizing denominators and simplifying complex expressions.
By using conjugates, you can elegantly work through expressions that initially seem complex. They simplify calculations and are a handy tool in many algebraic processes.
The FOIL Method
The FOIL method is a powerful technique for multiplying two binomials. It stands for First, Outer, Inner, Last, referring to which terms you should multiply together in the process.

Here鈥檚 how you can use FOIL:
  • **First**: Multiply the first terms in each binomial.
  • **Outer**: Multiply the outer terms in the product.
  • **Inner**: Multiply the inner terms.
  • **Last**: Multiply the last terms in each binomial.
Consider multiplying \((a + b)(a - b)\) using the FOIL method:
  • First: \(a \times a = a^2\)
  • Outer: \(a \times (-b) = -ab\)
  • Inner: \(b \times a = ab\)
  • Last: \(b \times (-b) = -b^2\)
Together, these give you \(a^2 - ab + ab - b^2\). The terms \(-ab + ab\) cancel each other out, reducing the expression to \(a^2 - b^2\).

The FOIL method not only guides you through multiplying binomials but also shows how conjugates simplify relationships, resulting in a more straightforward expression.

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Perform each division. Assume no division by \(0 .\) $$\frac{25 x^{2}-16}{5 x-4}$$

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