Simplify: $(3 x-4)(7 x+2)$
A. $10 x-2$
B. $21 x^2-22 x-8$
C. $21 x-8$
D. $21 x^2-5 x-8$
Answer (1)
Expand the product of the two binomials using the distributive property: ( 3 x − 4 ) ( 7 x + 2 ) = ( 3 x ) ( 7 x ) + ( 3 x ) ( 2 ) + ( − 4 ) ( 7 x ) + ( − 4 ) ( 2 ) .
Simplify each term: 21 x 2 + 6 x − 28 x − 8 .
Combine like terms: 6 x − 28 x = − 22 x .
The simplified expression is 21 x 2 − 22 x − 8 .
Explanation
Understanding the Problem We are given the expression ( 3 x − 4 ) ( 7 x + 2 ) and asked to simplify it. This involves expanding the product of two binomials.
Expanding the Product To simplify the expression, we will use the distributive property (also known as the FOIL method) to expand the product of the two binomials: ( 3 x − 4 ) ( 7 x + 2 ) = ( 3 x ) ( 7 x ) + ( 3 x ) ( 2 ) + ( − 4 ) ( 7 x ) + ( − 4 ) ( 2 )
Simplifying Each Term Now, we simplify each term: ( 3 x ) ( 7 x ) = 21 x 2 ( 3 x ) ( 2 ) = 6 x ( − 4 ) ( 7 x ) = − 28 x ( − 4 ) ( 2 ) = − 8 So, the expression becomes: 21 x 2 + 6 x − 28 x − 8
Combining Like Terms Next, we combine the like terms (the terms with the same power of x ): 6 x − 28 x = − 22 x So, the expression simplifies to: 21 x 2 − 22 x − 8
Final Answer Therefore, the simplified expression is 21 x 2 − 22 x − 8 .
Examples
Understanding how to simplify algebraic expressions like this is crucial in many areas, such as physics and engineering, where you often need to manipulate equations to solve for unknown variables. For example, if you're designing a bridge, you might use similar algebraic techniques to calculate the forces acting on different parts of the structure. Simplifying expressions allows engineers to optimize designs and ensure safety and efficiency. Also, in economics, you might use similar techniques to model supply and demand curves, helping to predict market behavior and make informed decisions.
