Problem 31 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{3}+4)(\sqrt{3}-7)\)

Step-by-Step Solution

Answer

-25 - 3√3

Step 1: Identify the given expression

We are given the expression \[ (\sqrt{3} + 4)(\sqrt{3} - 7). \]This resembles the multiplication of two binomials.

Step 2: Apply the distributive property

We will apply the distributive property (also known as the FOIL method for binomials) to find the product:- First, multiply the first terms: \( \sqrt{3} \times \sqrt{3} = 3 \).- Outer, multiply the outer terms: \( \sqrt{3} \times -7 = -7\sqrt{3} \).- Inner, multiply the inner terms: \( 4 \times \sqrt{3} = 4\sqrt{3} \).- Last, multiply the last terms: \( 4 \times -7 = -28 \).

Step 3: Combine like terms

Combine the results from Step 2:- Combine \( -7\sqrt{3} \) and \( 4\sqrt{3} \), which gives \( -3\sqrt{3} \).- Add the entire expression together:\[ 3 - 3\sqrt{3} - 28. \]

Step 4: Simplify the expression

Combine the constant terms (3 and -28):\[ 3 - 28 = -25. \]Therefore, the expression simplifies to:\[ -25 - 3\sqrt{3}. \]

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that allows us to multiply a single term by each term within a binomial. This technique is especially useful when multiplying binomials and is often referred to as the FOIL method for binomials, which stands for First, Outer, Inner, Last.

Here's how you apply it in practice:

• First: Multiply the first terms of each binomial together.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

In the given exercise, we applied each part of this method to the expression \[(\sqrt{3} + 4)(\sqrt{3} - 7)\] to break it down into simpler parts before combining the results.

Binomials

A binomial is a polynomial with exactly two terms. In the exercise, the expressions \(\sqrt{3} + 4\) and \(\sqrt{3} - 7\) are binomials. Understanding binomials is crucial because binomial multiplication is a common task in algebra which frequently leverages the distributive property.

When multiplying binomials, each term of the first binomial must be multiplied by each term of the second binomial. This ensures that all possible products are accounted for, which is essential for complete and accurate polynomial expansion.

The given example illustrates this method clearly, showing how each component of both binomials interact during multiplication.

Combining Like Terms

Combining like terms is an important simplification step. After applying the distributive property, you often end up with expressions containing similar terms that can be combined. Like terms are terms that contain the same variable raised to the same power.

In this exercise, after using the distributive property, we obtained terms like \(-7\sqrt{3}\) and \(4\sqrt{3}\). Both are considered 'like terms' because they contain the same radical expression \(\sqrt{3}\).

• We combined these by adding the coefficients together: \(-7 + 4 = -3\).
• This resulted in \(-3\sqrt{3}\).

Alongside simplifying constants, like combining \(3 - 28\) to get \(-25\), combining like terms refines the expression to its simplest form, \(-25 - 3\sqrt{3}\).