Which expression is equivalent to $\left(x^2-5 x\right)\left(3 x^2+2 x-9\right)$ ?
A. $x^2\left(3 x^2+2 x\right)-5 x(2 x-9)$
B. $x^2\left(3 x^2+2 x-9\right)-5 x$
C. $x^2\left(3 x^2+2 x-9\right)+5 x\left(3 x^2+2 x-9\right)$
D. $x^2\left(3 x^2+2 x-9\right)-5 x\left(3 x^2+2 x-9\right)$
Answer (1)
Distribute the terms in the expression ( x 2 − 5 x ) ( 3 x 2 + 2 x − 9 ) .
Apply the distributive property: x 2 ( 3 x 2 + 2 x − 9 ) − 5 x ( 3 x 2 + 2 x − 9 ) .
Compare the result with the given options.
The equivalent expression is D .
Explanation
Understanding the Problem We are given the expression ( x 2 − 5 x ) ( 3 x 2 + 2 x − 9 ) and asked to find an equivalent expression from the options provided. The key here is to recognize the distributive property.
Applying the Distributive Property The given expression is a product of two terms: ( x 2 − 5 x ) and ( 3 x 2 + 2 x − 9 ) . We can distribute the first term across the second term. This means we multiply each part of the first term by the entire second term.
Finding the Equivalent Expression So, we have: ( x 2 − 5 x ) ( 3 x 2 + 2 x − 9 ) = x 2 ( 3 x 2 + 2 x − 9 ) − 5 x ( 3 x 2 + 2 x − 9 ) This matches option D.
Verifying Other Options Let's examine the other options to confirm they are not equivalent: Option A: x 2 ( 3 x 2 + 2 x ) − 5 x ( 2 x − 9 ) = 3 x 4 + 2 x 3 − 10 x 2 + 45 x , which is not the same as the original expression. Option B: x 2 ( 3 x 2 + 2 x − 9 ) − 5 x = 3 x 4 + 2 x 3 − 9 x 2 − 5 x , which is also not the same as the original expression. Option C: x 2 ( 3 x 2 + 2 x − 9 ) + 5 x ( 3 x 2 + 2 x − 9 ) = 3 x 4 + 2 x 3 − 9 x 2 + 15 x 3 + 10 x 2 − 45 x = 3 x 4 + 17 x 3 + x 2 − 45 x , which is not the same as the original expression.
Final Answer Therefore, the equivalent expression is option D.
Examples
Understanding how to expand and simplify polynomial expressions like this is fundamental in many areas of mathematics and engineering. For example, when designing a bridge, engineers use polynomial equations to model the load and stress distribution. Simplifying these expressions allows them to accurately predict how the bridge will behave under different conditions, ensuring its safety and stability. Similarly, in economics, polynomial functions can model cost and revenue, and simplifying these expressions helps in optimizing business strategies.
