Problem 64 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (3 \sqrt{x}+2)(\sqrt{3 x}-2) $$

Step-by-Step Solution

Answer

The product is \(3x\sqrt{3} - 6\sqrt{x} + 2\sqrt{3x} - 4\).

Step 1: Distribute First Terms

Begin by distributing the first term of the first binomial, which is \( 3 \sqrt{x} \), over the second binomial \( (\sqrt{3x} - 2) \). Multiply \( 3 \sqrt{x} \times \sqrt{3x} \) and \( 3 \sqrt{x} \times -2 \):\[ 3 \sqrt{x} \times \sqrt{3x} = 3 \sqrt{x} \cdot \sqrt{3x} = 3 \sqrt{3x^2} = 3 \times x \sqrt{3} = 3x\sqrt{3} \]\[ 3 \sqrt{x} \times -2 = -6 \sqrt{x} \]

Step 2: Distribute Second Terms

Distribute the second term of the first binomial, which is \( 2 \), over the second binomial \( (\sqrt{3x} - 2) \). Multiply \( 2 \times \sqrt{3x} \) and \( 2 \times -2 \):\[ 2 \times \sqrt{3x} = 2 \sqrt{3x} \]\[ 2 \times -2 = -4 \]

Step 3: Combine Like Terms

Combine all the results from the distribution:\[ 3x \sqrt{3} - 6 \sqrt{x} + 2 \sqrt{3x} - 4 \]Note that there are no like terms to combine in this expression, so this is the simplified polynomial.

Key Concepts

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial expression to its simplest form. This involves several steps like eliminating like terms, reducing fractions, and ensuring there are no unnecessary terms. In our exercise \((3 \sqrt{x}+2)(\sqrt{3 x}-2)\), we deal with a product of two binomials. After expansion by distribution, we get the expression:

• 3x \sqrt{3} - 6 \sqrt{x} + 2 \sqrt{3x} - 4

This expression is in its simplest form because:

• There are no like terms left to combine.
• The terms are arranged systematically.

Simplifying polynomials helps in making complex algebraic equations manageable and easier to solve. Ensuring each term is distinct allows for a clear understanding and analysis of the expression.

Distribution in Algebra

Distribution in algebra plays a crucial role in multiplying expressions, particularly when dealing with binomials. In the original exercise \((3 \sqrt{x}+2)(\sqrt{3 x}-2)\), we use distribution to expand the expression. Here’s a step-by-step breakdown of the distribution process:

• Multiply the first term of the first binomial by each term of the second binomial.
• Repeat the above step with the second term of the first binomial.
• The results are:\(3 \sqrt{x} \cdot \sqrt{3x} = 3x \sqrt{3}\) and \(3 \sqrt{x} \cdot -2 = -6 \sqrt{x}\).
• Then, \(2 \cdot \sqrt{3x} = 2 \sqrt{3x}\) and \(2 \cdot -2 = -4\).

Each product from the distribution process contributes to forming the complete polynomial after combining the resulting terms. This method helps efficiently manage and solve various algebraic expressions, especially when dealing with polynomials.

Radical Expressions

Radical expressions involve roots, such as square roots, of variables or numbers. They can be tricky to handle if not approached correctly. The original problem includes terms like \(\sqrt{x}\) and \(\sqrt{3x}\). Here’s how to manage and understand them:

• When multiplying radicals with the same index, combine them under a single radical. For instance, \(\sqrt{x} \cdot \sqrt{3x} = \sqrt{3x^2}\), which simplifies to \(x\sqrt{3}\).
• Radicals can be distributed over addition and subtraction, aiding in simplification.
• Always simplify the radicand (the expression under the radical) as much as possible.

Understanding how to manipulate radical expressions is essential for simplifying or expanding polynomials that involve roots. It allows solving complicated expressions by dealing with familiar numerical values and providing clear, simplified results.