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Chapter 8: Problem 53

Simplify. $$(x-\sqrt{2})(x-\sqrt{3})$$

Short Answer

Expert verified
\( x^2 - x\sqrt{3} - x\sqrt{2} + \sqrt{6} \)

Step by step solution

01

- Multiply First Terms

Multiply the first terms from each binomial: \(x \times x = x^2\).
02

- Multiply Outer Terms

Multiply the outer terms from each binomial: \(x \times -\sqrt{3} = -x\sqrt{3}\).
03

- Multiply Inner Terms

Multiply the inner terms from each binomial: \(-\sqrt{2} \times x = -x\sqrt{2}\).
04

- Multiply Last Terms

Multiply the last terms from each binomial: \(-\sqrt{2} \times -\sqrt{3} = \sqrt{6}\).
05

- Combine Like Terms

Combine all the terms: \(x^2 - x\sqrt{3} - x\sqrt{2} + \sqrt{6}\).

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Key Concepts

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

Binomial Expansion
In algebra, binomial expansion involves multiplying two binomials to get a polynomial. A binomial has two terms, such as \((a + b)\) or \((x - \sqrt{2})\). When we multiply two binomials, we use the distributive property. This means we multiply each term in the first binomial by each term in the second binomial. For example, with \((x - \sqrt{2})(x - \sqrt{3})\):
  • First, we multiply the first terms: \(x \times x = x^2\).
  • Then, the outer terms: \(x \times -\sqrt{3} = -x\sqrt{3}\).
  • Next, the inner terms: \(-\sqrt{2} \times x = -x\sqrt{2}\).
  • Finally, the last terms: \(-\sqrt{2} \times -\sqrt{3} = \sqrt{6}\).
This makes the process easier and systematic, leading us to the expanded form.
Simplifying Expressions
After expanding a binomial expression, the next step is to simplify it by combining like terms. Like terms are terms that contain the same variables raised to the same power. For instance, \(2x \sqrt{3}\) and \(-x \sqrt{3}\) are like terms but \(x^2\) and \(-x \sqrt{2}\) are not.
In our initial expansion, we got \(x^2 - x\sqrt{3} - x\sqrt{2} + \sqrt{6}\). Here, \(x^2\) is one unique term, while \(-x\sqrt{3}\) and \(-x\sqrt{2}\) need no further simplification because they have different radicals. Therefore, our final step is simply to write them together in a single expression.
Algebraic Multiplication
Algebraic multiplication is one of the core principles to master in algebra. It includes multiplying variables and constants following specific rules. When multiplying algebraic terms, you follow the distributive property as we did above. This ensures all terms are accounted for. Consider these basics:
  • Multiplying similar bases: \(x \times x = x^2\).
  • Distributing across parentheses: \(a(b + c) = ab + ac\).
  • Handling negative signs: Negative \( \times\) Negative = Positive.
In our case, properly following these principles helps us efficiently expand and simplify \((x - \sqrt{2})(x - \sqrt{3})\). By remembering these steps, any complicated multiplication becomes manageable.

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