Multiply.
$(3 a b-1)(5 a b-6) =$
Answer (1)
Multiply the binomials using the distributive property (FOIL method).
Calculate each term: ( 3 ab ) ( 5 ab ) = 15 a 2 b 2 , ( 3 ab ) ( − 6 ) = − 18 ab , ( − 1 ) ( 5 ab ) = − 5 ab , ( − 1 ) ( − 6 ) = 6 .
Combine like terms: − 18 ab − 5 ab = − 23 ab .
Write the final expression: 15 a 2 b 2 − 23 ab + 6 .
Explanation
Understanding the problem We are asked to multiply two binomials: ( 3 ab − 1 ) and ( 5 ab − 6 ) . We will use the distributive property (also known as the FOIL method) to perform this multiplication.
Applying the distributive property Using the distributive property (FOIL method), we have: ( 3 ab − 1 ) ( 5 ab − 6 ) = ( 3 ab ) ( 5 ab ) + ( 3 ab ) ( − 6 ) + ( − 1 ) ( 5 ab ) + ( − 1 ) ( − 6 ) Each term is calculated as follows:
Calculating each term ( 3 ab ) ( 5 ab ) = 15 a 2 b 2 ( 3 ab ) ( − 6 ) = − 18 ab ( − 1 ) ( 5 ab ) = − 5 ab ( − 1 ) ( − 6 ) = 6
Combining like terms Now, we combine the like terms: − 18 ab − 5 ab = − 23 ab
Final expression Finally, we write the complete expression: 15 a 2 b 2 − 23 ab + 6 So, the product of ( 3 ab − 1 ) ( 5 ab − 6 ) is 15 a 2 b 2 − 23 ab + 6 .
Examples
Understanding how to multiply binomials like ( 3 ab − 1 ) ( 5 ab − 6 ) is fundamental in algebra and has practical applications. For instance, if you're calculating the area of a rectangular garden where the sides are expressed as binomials involving variables, this multiplication technique helps you find the total area. Similarly, in physics, when dealing with equations of motion or energy that involve products of binomials, this method simplifies the expressions and aids in problem-solving. Mastering binomial multiplication enhances your ability to model and solve real-world problems across various disciplines.
