Multiply $(2 x+2)(3 x-1)$
a. $6 x^2+4 x-2$
b. $5 x^2+3 x-2$
c. $5 x^2+4 x-2$
d. $6 x^2+3 x+1$
Answer (1)
Use the distributive property (FOIL method) to expand the product: ( 2 x + 2 ) ( 3 x − 1 ) = ( 2 x ) ( 3 x ) + ( 2 x ) ( − 1 ) + ( 2 ) ( 3 x ) + ( 2 ) ( − 1 ) .
Simplify each term: 6 x 2 − 2 x + 6 x − 2 .
Combine like terms: 6 x 2 + 4 x − 2 .
The final answer is: 6 x 2 + 4 x − 2 .
Explanation
Understanding the Problem We are given the expression ( 2 x + 2 ) ( 3 x − 1 ) and asked to multiply it. This involves expanding the product of two binomials.
Applying the Distributive Property To multiply the expression ( 2 x + 2 ) ( 3 x − 1 ) , we use the distributive property (also known as the FOIL method). This means we multiply each term in the first binomial by each term in the second binomial: ( 2 x + 2 ) ( 3 x − 1 ) = ( 2 x ) ( 3 x ) + ( 2 x ) ( − 1 ) + ( 2 ) ( 3 x ) + ( 2 ) ( − 1 )
Simplifying the Expression Now, we simplify each term: ( 2 x ) ( 3 x ) = 6 x 2 ( 2 x ) ( − 1 ) = − 2 x ( 2 ) ( 3 x ) = 6 x ( 2 ) ( − 1 ) = − 2 So, the expression becomes: 6 x 2 − 2 x + 6 x − 2
Combining Like Terms Next, we combine like terms (the terms with the same power of x ): − 2 x + 6 x = 4 x So, the expression simplifies to: 6 x 2 + 4 x − 2
Identifying the Correct Option Finally, we compare our result with the given options. The correct answer is: 6 x 2 + 4 x − 2 which corresponds to option a.
Examples
Understanding how to multiply binomials is essential in various fields, such as physics and engineering, where polynomial expressions are used to model physical phenomena. For example, when calculating the area of a rectangular garden with sides ( 2 x + 2 ) and ( 3 x − 1 ) , multiplying these binomials gives the area as a quadratic expression. This skill is also crucial in economics for modeling cost and revenue functions, helping businesses optimize their operations and make informed decisions.
