How to Multiply Binomial Radical Expressions With Variables

After multiplying binomials with radicals, you can check your answer with a calculator.

Binomials are algebraic expressions with variables that contain two terms -- basically chunks separated by addition or subtraction signs. To simplify or work with algebraic equations, you need to know how to multiply two binomials together using the first, outer, inner, last, or FOIL, method. Binomials that contain radicals, or roots such as square roots, can be a little trickier. However, if you understand how to perform operations with radicals, you should not have difficulty learning the process.

Instructions

1. Copy the multiplication problem from the textbook or worksheet. Each binomial should be in parenthesis like this: (2x - 3 sqrt6x)(5x + 2sqrt3x).

2. Locate the first term of the binomial to the left (in the example from Step 1, it would be 2x) and multiply it with the first term in the second binomial (in this case 5x). Add the exponents together to get the new exponent. Here, 2x multiplied by 5x would give you 10x^2. Write the answer on the line beneath the problem as the first term.

3. Multiply the first term of the binomial on the left with the second term of the binomial on the right, and write the answer as the second term on the line below. If radicals are involved, multiply together any numbers or variables on the outside of the square root signs and leave them on the outside. Multiply, and apply exponent rules when necessary, any variables and numbers on the inside of the square root signs. Leave these under the radical. For example, 2x and 2sqrt3x would become 4xsqrt3x.

4. Multiply the second term of the left binomial with the first term from the right. Fill in the answer as the third term on the line below. Here, - 3 sqrt6x and 5x would become -15sqrt-30x.

5. Multiply the second term of the left binomial with the second term of the one on the right, and use the answer as the final term on the next line. In this case, -3sqrt6x would multiply with 2sqrt3x to give you -6sqrt(18x^2).

6. Simplify any radicals that you can in your new expression. For instance, -6sqrt(18x^2) would simplify to -18xsqrt2.

7. Combine any like terms. For two terms to be able to combine, they have to have exactly the same variables to the same powers and the same radicals. For example, 3xsqrt5 + 8xsqrt5 would simplify to 11xsqrt5. However, 3xsqrt5 + 8xsqrt4 cannot be simplified further.

Tips & Warnings

• Remember to think about signs when multiplying terms together. Therefore, if you are multiplying -3x by 4x, you would get -12x^2.

• Check your answer by substituting real numbers for the variables in both the answers and the original. Then perform the operations on the calculator and see if the numbers match.

References

• The Math Page: Multiplying Radicals in Polynomials
• Regents Prep: Simplifying Radicals