Problem 35 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (\sqrt{2}+1)(\sqrt{2}-3) $$

Step-by-Step Solution

Answer

The simplified expression is \(-1 - 2\sqrt{2}\).

Step 1: Distribute Terms

To multiply the two binomials, we use the distributive property, also known as the FOIL method (First, Outer, Inner, Last). We distribute each term in the first binomial by each term in the second binomial:First: \( \sqrt{2} \times \sqrt{2} = (\sqrt{2})^2 = 2 \)Outer: \( \sqrt{2} \times (-3) = -3\sqrt{2} \)Inner: \( 1 \times \sqrt{2} = \sqrt{2} \)Last: \( 1 \times (-3) = -3 \).

Step 2: Combine Like Terms

After distributing, we combine all the terms:\( 2 - 3\sqrt{2} + \sqrt{2} - 3 \).Combine the like terms:- For the constants: \( 2 - 3 = -1 \).- For the square root terms: \(-3\sqrt{2} + \sqrt{2} = -2\sqrt{2} \).Putting these together, we have:\( -1 - 2\sqrt{2} \).

Step 3: Final Simplification

The resulting expression \(-1 - 2\sqrt{2}\) is already simplified because all like terms have been combined, and there are no further calculations or reductions to perform.

Key Concepts

Multiplying Binomials

Multplying binomials may initially look complex, especially when dealing with square roots, but is made simple using the FOIL method. FOIL stands for First, Outer, Inner, and Last, which refers to the order of multiplying terms.

• First: Multiply the first term of each binomial. In our exercise, it's \( \sqrt{2} \times \sqrt{2} \), which becomes \( (\sqrt{2})^2 = 2 \).


• Outer: Multiply the outer terms: \( \sqrt{2} \times (-3) = -3\sqrt{2} \).


• Inner: Multiply the inner terms: \( 1 \times \sqrt{2} = \sqrt{2} \).


• Last: Multiply the last term of each binomial: \( 1 \times (-3) = -3 \).

Looking closely, you'll see that each term ensures none of the components of either binomial is missed during multiplication. This thoroughness helps capture the full product of the binomials.

Simplification

Once we've multiplied the binomials, the resulting expression \( 2 - 3\sqrt{2} + \sqrt{2} - 3 \) might seem cluttered. Simplification is key to understand and manage such expressions effectively.

• Combine constants: The constants here are 2 and -3. Combine them to get -1. This forms the straightforward part of the expression.


• Combine like terms: Look for terms sharing the same radical. Here, we have \(-3\sqrt{2}\) and \(+\sqrt{2}\). Combining these yields \(-2\sqrt{2}\).

Finally, the simplified expression \(-1 - 2\sqrt{2}\) is easier to interpret and much neater, making further operations, if necessary, less prone to errors.

Distributive Property

The distributive property, a key concept in algebra, allows us to multiply a single term by each term within a parenthesis. It is the backbone for methods like FOIL that handle binomials.In our example, we use the property to ensure each part of the first binomial ((\(\sqrt{2}\) and 1) gets multiplied by every part of the second binomial \((-3)\sqrt{2}-3)\). By understanding this property:

• You're able to expand expressions methodically and precisely.


• You'll grasp deeper insights into how algebraic expressions behave when combined.

Mastering the distributive property is crucial for tackling more complex algebraic expressions where manipulation and expansion are required.