Which expression is equivalent to $(3 x^2+4 x-7)(x-2)$?
A. $(3 x^2+4 x-7)+2(3 x^2+4 x-7)$
B. $2 x(3 x^2+4 x-7)$
C. $(3 x^2+4 x-7)(x)+(3 x^2+4 x-7)(-2)$
D. $x(3 x^2+4 x-7)-2
Answer (2)
Expand the given expression: ( 3 x 2 + 4 x − 7 ) ( x − 2 ) = 3 x 3 − 2 x 2 − 15 x + 14 .
Expand option A: ( 3 x 2 + 4 x − 7 ) + 2 ( 3 x 2 + 4 x − 7 ) = 9 x 2 + 12 x − 21 .
Expand option B: 2 x ( 3 x 2 + 4 x − 7 ) = 6 x 3 + 8 x 2 − 14 x .
Option C: ( 3 x 2 + 4 x − 7 ) ( x ) + ( 3 x 2 + 4 x − 7 ) ( − 2 ) is equivalent to the given expression.
Expand option D: x ( 3 x 2 + 4 x − 7 ) − 2 = 3 x 3 + 4 x 2 − 7 x − 2 .
The equivalent expression is C .
Explanation
Understanding the Problem We are given the expression ( 3 x 2 + 4 x − 7 ) ( x − 2 ) and four other expressions. Our goal is to find the expression among the four options that is equivalent to the given expression. To do this, we will expand the given expression and compare it to the expanded forms of the options.
Expanding the Given Expression Let's expand the given expression ( 3 x 2 + 4 x − 7 ) ( x − 2 ) using the distributive property:
( 3 x 2 + 4 x − 7 ) ( x − 2 ) = 3 x 2 ( x − 2 ) + 4 x ( x − 2 ) − 7 ( x − 2 )
= 3 x 3 − 6 x 2 + 4 x 2 − 8 x − 7 x + 14
= 3 x 3 − 2 x 2 − 15 x + 14
Analyzing the Options Now, let's examine the options:
Option A: ( 3 x 2 + 4 x − 7 ) + 2 ( 3 x 2 + 4 x − 7 ) = 3 ( 3 x 2 + 4 x − 7 ) = 9 x 2 + 12 x − 21 . This is not equal to 3 x 3 − 2 x 2 − 15 x + 14 .
Option B: 2 x ( 3 x 2 + 4 x − 7 ) = 6 x 3 + 8 x 2 − 14 x . This is not equal to 3 x 3 − 2 x 2 − 15 x + 14 .
Option C: ( 3 x 2 + 4 x − 7 ) ( x ) + ( 3 x 2 + 4 x − 7 ) ( − 2 ) = ( 3 x 2 + 4 x − 7 ) ( x − 2 ) . This is exactly the expression we started with, so it must be equivalent.
Option D: x ( 3 x 2 + 4 x − 7 ) − 2 = 3 x 3 + 4 x 2 − 7 x − 2 . This is not equal to 3 x 3 − 2 x 2 − 15 x + 14 .
Final Answer Therefore, the expression equivalent to ( 3 x 2 + 4 x − 7 ) ( x − 2 ) is option C: ( 3 x 2 + 4 x − 7 ) ( x ) + ( 3 x 2 + 4 x − 7 ) ( − 2 ) .
Examples
Understanding polynomial multiplication is crucial in various fields, such as engineering and computer science. For instance, when designing a bridge, engineers use polynomial equations to model the load distribution and structural integrity. Similarly, in computer graphics, polynomials are used to create smooth curves and surfaces. By mastering polynomial multiplication, you can tackle real-world problems involving complex relationships between variables, ensuring accuracy and efficiency in your calculations.
The expression equivalent to ( 3 x 2 + 4 x − 7 ) ( x − 2 ) is Option C: ( 3 x 2 + 4 x − 7 ) ( x ) + ( 3 x 2 + 4 x − 7 ) ( − 2 ) . This is justified by comparing the expansions of all options. Only Option C matches the original expression.
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