Multiply $(x-4)(2 x+3)$ using the distributive property. Select the answer choice showing the correct distribution.
A. $(x-4)(2 x)+(2 x)(3)$
B. $(x)(2 x+3)+(x-4)$
C. $(x-4)(2 x)+(x-4)(3)$
D. $(x)(2 x)+3(x)+2 x+3$
Answer (1)
The problem requires applying the distributive property to the expression ( x − 4 ) ( 2 x + 3 ) .
Distribute ( x − 4 ) over ( 2 x + 3 ) to get ( x − 4 ) ( 2 x ) + ( x − 4 ) ( 3 ) .
Compare the result with the given options.
Identify the correct option that matches the distributive property.
The correct answer is C .
Explanation
Understanding the Problem We are asked to multiply ( x − 4 ) ( 2 x + 3 ) using the distributive property and to identify the correct application of the distributive property from the given options.
Understanding the Distributive Property The distributive property states that a ( b + c ) = ab + a c . We need to apply this property to the expression ( x − 4 ) ( 2 x + 3 ) . We can think of ( x − 4 ) as a single term and distribute it over ( 2 x + 3 ) , or we can think of ( 2 x + 3 ) as a single term and distribute it over ( x − 4 ) .
Applying the Distributive Property Applying the distributive property, we can distribute ( x − 4 ) over ( 2 x + 3 ) as follows: ( x − 4 ) ( 2 x + 3 ) = ( x − 4 ) ( 2 x ) + ( x − 4 ) ( 3 ) This means we multiply ( x − 4 ) by 2 x and then multiply ( x − 4 ) by 3 , and add the results.
Comparing with the Options Now, let's compare our result with the given options:
A. ( x − 4 ) ( 2 x ) + ( 2 x ) ( 3 ) - This is incorrect because it distributes 2 x over ( x − 4 ) in the second term, which is not what we want.
B. ( x ) ( 2 x + 3 ) + ( x − 4 ) - This is incorrect because it only distributes x over ( 2 x + 3 ) and then adds ( x − 4 ) , which is not the correct application of the distributive property.
C. ( x − 4 ) ( 2 x ) + ( x − 4 ) ( 3 ) - This is the correct application of the distributive property, as we derived above.
D. ( x ) ( 2 x ) + 3 ( x ) + 2 x + 3 - This is incorrect because it seems to be expanding the expression, but it doesn't show the initial application of the distributive property.
Final Answer Therefore, the correct answer is C. ( x − 4 ) ( 2 x ) + ( x − 4 ) ( 3 ) .
Examples
The distributive property is a fundamental concept in algebra and is used in various real-life scenarios. For example, suppose you are buying 5 items that cost x dollars each and also have to pay a fixed shipping fee of $3. The total cost can be represented as 5 ( x + 3 ) . Using the distributive property, you can calculate the total cost as 5 x + 15 , where 5 x is the cost of the items and $15 is the total shipping fee.
