Multiply Binomial Radical Expressions - Algebra-2

1. Fundamental Concepts

Binomial Radical Expression: A radical expression with two terms. The terms can be in the form of "rational term + radical term" or "radical term + radical term", such as \((3 + \sqrt{2})\) and \((\sqrt{5} - 2\sqrt{3})\).
Conjugate of a Binomial Radical: For a binomial radical of the form \((a + b\sqrt{c})\), its conjugate is \((a - b\sqrt{c})\) (only the sign between the two terms is changed), and vice versa. For example, the conjugate of \((2 + \sqrt{7})\) is \((2 - \sqrt{7})\), and the conjugate of \((\sqrt{3} - 4\sqrt{2})\) is \((\sqrt{3} + 4\sqrt{2})\).
FOIL Method: A standard method for multiplying two binomials, which stands for "First (multiply the first terms) → Outer (multiply the outer terms) → Inner (multiply the inner terms) → Last (multiply the last terms)". Finally, add up all the results.

2. Key Concepts

Essence of Multiplying Binomial Radicals: Conversion to Polynomial Multiplication Multiplying binomial radicals follows the same rule as multiplying ordinary binomials (e.g., \((x + 2)(x - 3)\)). It needs to be expanded using the FOIL method or the distributive property, and then like terms (terms containing like radicals) are combined.
Special Property of Multiplying Conjugates: Result is a Rational Expression When multiplying conjugate binomial radicals, the products of the outer and inner terms are opposites of each other; they cancel out when added. The final result is "the square of the first term minus the square of the last term", which is expressed as:\((a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - (b\sqrt{c})^2 = a^2 - b^2c\) (the result has no radicals and is a rational number or integral expression).
Simplification Required After Expansion: Combine Like Radicals + Arrange Coefficients After expanding the product of binomial radicals, add the coefficients of like radicals (e.g., \(3\sqrt{2}\) and \(5\sqrt{2}\)), and ensure all radicals are in their simplified form (e.g., \(\sqrt{12}\) should be converted to \(2\sqrt{3}\)).

3. Examples

1. Easy Difficulty (Multiplying Binomial Radicals Without Variables)

Question: Calculate \((2 + \sqrt{3})(4 - \sqrt{3})\)
Solution:
Step 1: Expand using the FOIL method:

First terms: \(2 \times 4 = 8\);

Outer terms: \(2 \times (-\sqrt{3}) = -2\sqrt{3}\);

Inner terms: \(\sqrt{3} \times 4 = 4\sqrt{3}\);

Last terms: \(\sqrt{3} \times (-\sqrt{3}) = -(\sqrt{3})^2 = -3\).
Step 2: Combine like terms and constant terms:\(8 - 2\sqrt{3} + 4\sqrt{3} - 3 = (8 - 3) + (-2\sqrt{3} + 4\sqrt{3}) = 5 + 2\sqrt{3}\).

2. Medium Difficulty (Multiplying Binomial Radicals With Variables)

Question: Calculate \((x + 2\sqrt{x})(x - \sqrt{x})\)（\(x \geq 0\)）
Solution:
Step 1: Expand using the FOIL method:

First terms: \(x \times x = x^2\);

Outer terms: \(x \times (-\sqrt{x}) = -x\sqrt{x}\);

Inner terms: \(2\sqrt{x} \times x = 2x\sqrt{x}\);

Last terms: \(2\sqrt{x} \times (-\sqrt{x}) = -2(\sqrt{x})^2 = -2x\).
Step 2: Combine like radicals and like terms:\(x^2 - x\sqrt{x} + 2x\sqrt{x} - 2x = x^2 + (-x\sqrt{x} + 2x\sqrt{x}) - 2x = x^2 + x\sqrt{x} - 2x\).

3. Difficult Difficulty (Combining Conjugates and Multi-Step Simplification)

Question: Calculate \((3\sqrt{2} - \sqrt{5})(3\sqrt{2} + \sqrt{5}) + \sqrt{3}(\sqrt{6} - 2\sqrt{3})\)
Solution:
Step 1: First calculate the product of the conjugates (using the difference of squares formula):\((3\sqrt{2} - \sqrt{5})(3\sqrt{2} + \sqrt{5}) = (3\sqrt{2})^2 - (\sqrt{5})^2 = 9 \times 2 - 5 = 18 - 5 = 13\).
Step 2: Calculate the product of the monomial and the binomial radical:\(\sqrt{3}(\sqrt{6} - 2\sqrt{3}) = \sqrt{3 \times 6} - 2 \times (\sqrt{3})^2 = \sqrt{18} - 2 \times 3 = 3\sqrt{2} - 6\) (simplify \(\sqrt{18}\) to \(3\sqrt{2}\)).
Step 3: Combine the results of the two parts:\(13 + 3\sqrt{2} - 6 = (13 - 6) + 3\sqrt{2} = 7 + 3\sqrt{2}\).

4. Problem-Solving Techniques

Remember the FOIL Method and Avoid Missing Terms When expanding, strictly follow the order of "First → Outer → Inner → Last" to multiply, ensuring that no one of the four product terms is missing (pay special attention to signs, e.g., when outer or inner terms have negative signs).
Prioritize the Difference of Squares Formula for Conjugates to Simplify Calculations If multiplying conjugate binomial radicals, directly apply \((a + b)(a - b) = a^2 - b^2\) instead of expanding completely, which reduces the number of calculation steps (as in Step 1 of the difficult difficulty example).
Simplify Step by Step: Calculate Products First, Then Combine If the expression contains multiple sets of multiplications (e.g., "product of conjugates + product of a monomial and a binomial radical"), calculate and simplify each set of products separately first, then combine the final results to avoid confusion.
Check the Simplification of Radicals and the Correctness of Signs After completing the calculation, verify two points: ① Whether all radicals are in their simplified form (e.g., \(\sqrt{20}\) should be converted to \(2\sqrt{5}\)); ② Whether the signs are correct (especially the handling of negative signs in outer terms, inner terms, and last terms).