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Chapter 10: Problem 21

How are the problems Multiply \((x+8)(x+3)\) and Multiply \((3+\sqrt{2})(1+\sqrt{2})\) similar? What method can be used to multiply each of them?

Short Answer

Expert verified
Both problems involve the multiplication of two binomials and can be solved using the FOIL method. By applying the FOIL method, we can systematically multiply and combine the terms from both binomials to obtain the final simplified expressions: \(x^2+11x+24\) for \((x+8)(x+3)\) and \(5+4\sqrt{2}\) for \((3+\sqrt{2})(1+\sqrt{2})\).

Step by step solution

01

Identify the Form of Both Expressions

Both problems involve the multiplication of two binomials. A binomial is an algebraic expression that has two terms. In the first case, we have \((x+8)(x+3)\), where \(x+8\) and \(x+3\) are binomials. In the second case, we have \((3+\sqrt{2})(1+\sqrt{2})\), where \(3+\sqrt{2}\) and \(1+\sqrt{2}\) are binomials.
02

Recognize the Multiplication Method

The method that can be used to multiply these binomials is called the "FOIL method." FOIL stands for First, Outer, Inner, and Last, referring to the terms within the binomials that need to be multiplied. We can apply this method to both expressions.
03

Apply the FOIL Method to the First Expression

We first apply the FOIL method to \((x+8)(x+3)\): 1. First: Multiply the first terms of both binomials: \(x \cdot x = x^2\) 2. Outer: Multiply the outer terms of both binomials: \(x \cdot 3 = 3x\) 3. Inner: Multiply the inner terms of both binomials: \(8 \cdot x = 8x\) 4. Last: Multiply the last terms of both binomials: \(8 \cdot 3 = 24\) Now combine the results of each multiplication: \(x^2 + 3x + 8x + 24\) which simplifies to \(x^2 + 11x + 24\).
04

Apply the FOIL Method to the Second Expression

We now apply the FOIL method to \((3+\sqrt{2})(1+\sqrt{2})\): 1. First: Multiply the first terms of both binomials: \(3 \cdot 1 = 3\) 2. Outer: Multiply the outer terms of both binomials: \(3 \cdot \sqrt{2} = 3\sqrt{2}\) 3. Inner: Multiply the inner terms of both binomials: \(\sqrt{2} \cdot 1 = \sqrt{2}\) 4. Last: Multiply the last terms of both binomials: \(\sqrt{2} \cdot \sqrt{2} = 2\) Now combine the results of each multiplication: \(3 + 3\sqrt{2} + \sqrt{2} + 2\) which simplifies to \(5 + 4\sqrt{2}\).
05

Conclusion

The similarities between the two given problems are that both involve multiplication of two binomials and can be solved using the FOIL method. The FOIL method allows us to systematically multiply and combine the terms from both binomials to obtain the final simplified expression.

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Most popular questions from this chapter

In your own words, explain how to rationalize the denominator of an expression containing one term in the denominator.

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