Problem 33 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{5}-6)(\sqrt{5}-3)\)

Step-by-Step Solution

Answer

The product is \(23 - 9\sqrt{5}\) in simplest radical form.

Step 1: Identify the Expression

The given expression is \((\sqrt{5} - 6)(\sqrt{5} - 3)\). We need to find the product of these two binomials.

Step 2: Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, and Last. It is used to multiply two binomials. Begin by multiplying the first terms, \(\sqrt{5} \cdot \sqrt{5}\), which equals 5.

Step 3: Multiply the Outer Terms

Next, multiply the outer terms: \(\sqrt{5} \cdot -3\), which simplifies to \(-3\sqrt{5}\).

Step 4: Multiply the Inner Terms

Now, multiply the inner terms: \(-6 \cdot \sqrt{5}\), which gives \(-6\sqrt{5}\).

Step 5: Multiply the Last Terms

Finally, multiply the last terms: \(-6 \cdot -3\), which equals 18.

Step 6: Combine Like Terms

Combine the terms you obtained from the FOIL method: - First term: 5- Outer and Inner terms: \(-3\sqrt{5}\) and \(-6\sqrt{5}\) combine to \(-9\sqrt{5}\)- Last term: 18Add them together: \(5 - 9\sqrt{5} + 18\).

Step 7: Simplify the Expression

Combine the constant terms 5 and 18 to get 23. The simplified form of the expression is \(23 - 9\sqrt{5}\).

Key Concepts

FOIL Method

The FOIL method is a handy technique to multiply two binomials. Its name is an acronym that helps you remember the steps: First, Outer, Inner, and Last. Each of these terms represents a pair of terms in the binomials that need to be multiplied together. Let's break it down.

• First: Multiply the first terms in each binomial. In our expression \((\sqrt{5} - 6)(\sqrt{5} - 3)\), it means multiplying \(\sqrt{5} \cdot \sqrt{5}\) which equals 5.
• Outer: Multiply the outermost terms in the binomial pair. This means multiplying \(\sqrt{5} \cdot -3\), giving us \(-3\sqrt{5}\).
• Inner: Multiply the inner terms of the binomials. Here, \(-6\cdot\sqrt{5}\) results in \(-6\sqrt{5}\).
• Last: Multiply the last terms in each binomial, producing \(-6 \cdot -3\), which is 18.

Following the FOIL method ensures that you account for all possible products of pairs from the two binomials, leading to a complete product expression.

Binomials

Binomials are expressions consisting of two terms separated by a plus or minus sign. Understanding them is crucial for operations like multiplication, as demonstrated in our expression \((\sqrt{5} - 6)(\sqrt{5} - 3)\). Here, each binomial consists of a radical term and a constant.

• In the first binomial \((\sqrt{5} - 6)\), \(\sqrt{5}\) is a radical term and 6 is the constant.
• In the second binomial \((\sqrt{5} - 3)\), \(\sqrt{5}\) is again a radical, and 3 is another constant.

Multiplying binomials using methods like FOIL allows us to understand the behavior and outcome of combining such expressions. Binomials can combine like terms and expand into simpler forms.

Multiplying Radicals

Multiplying radicals involves dealing with numbers or variables under the root sign (√). When multiplying, you consider the numbers both inside and outside the radical.

• For instance, multiplying \(\sqrt{5} \cdot \sqrt{5}\) gives 5 because the square root signs cancel out when the numbers are the same.
• However, when multiplying expressions like \(\sqrt{5} \cdot -3\), you multiply the number outside by the radical itself, resulting in \(-3\sqrt{5}\).

Understanding how to multiply radicals is key to simplifying expressions and finding the simplest radical form, especially when combining like terms.

Like Terms

Like terms are components in an expression that have the same radical or variable, making them combinable. In our context, it refers to the combination of terms involving radicals.

• In the expression \(5 - 9\sqrt{5} + 18\), \(-9\sqrt{5}\) is the only term involving the radical \(\sqrt{5}\).
• You combine like terms by summing the coefficients while keeping the identical radical intact, simplifying the expression.
• For non-radical numbers, such as 5 and 18, add those directly to simplify further, resulting in 23.

By combining like terms, you make expressions easier to read and work with, often resulting in a simpler form like \(23 - 9\sqrt{5}\).