What is the product?
$\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)$
Answer (2)
To find the product of the given expression, we first multiply 7 x 2 with ( 2 x 3 + 5 ) to get 14 x 5 + 35 x 2 . We then multiply this result with ( x 2 − 4 x − 9 ) , leading to the final expanded product of 14 x 7 − 56 x 6 − 126 x 5 + 35 x 4 − 140 x 3 − 315 x 2 .
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Multiply ( 7 x 2 ) and ( 2 x 3 + 5 ) to get 14 x 5 + 35 x 2 .
Multiply the result ( 14 x 5 + 35 x 2 ) by ( x 2 − 4 x − 9 ) .
Expand the expression to obtain 14 x 7 − 56 x 6 − 126 x 5 + 35 x 4 − 140 x 3 − 315 x 2 .
The final product is 14 x 7 − 56 x 6 − 126 x 5 + 35 x 4 − 140 x 3 − 315 x 2 .
Explanation
Understanding the Problem We are given the expression ( 7 x 2 ) ( 2 x 3 + 5 ) ( x 2 − 4 x − 9 ) and asked to find the product of these polynomials.
Multiplying the First Two Terms First, we multiply ( 7 x 2 ) and ( 2 x 3 + 5 ) . This gives
( 7 x 2 ) ( 2 x 3 + 5 ) = 7 x 2 ⋅ 2 x 3 + 7 x 2 ⋅ 5 = 14 x 5 + 35 x 2 .
Multiplying by the Third Term Next, we multiply the result ( 14 x 5 + 35 x 2 ) with ( x 2 − 4 x − 9 ) . This gives
( 14 x 5 + 35 x 2 ) ( x 2 − 4 x − 9 ) = 14 x 5 ( x 2 − 4 x − 9 ) + 35 x 2 ( x 2 − 4 x − 9 ) .
Expanding the Expression Now, we expand the expression:
14 x 5 ( x 2 − 4 x − 9 ) + 35 x 2 ( x 2 − 4 x − 9 ) = 14 x 7 − 56 x 6 − 126 x 5 + 35 x 4 − 140 x 3 − 315 x 2 = 14 x 7 − 56 x 6 − 126 x 5 + 35 x 4 − 140 x 3 − 315 x 2 .
Final Result Therefore, the final result is 14 x 7 − 56 x 6 − 126 x 5 + 35 x 4 − 140 x 3 − 315 x 2 .
Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, in control systems, the transfer function of a system can be represented as a ratio of two polynomials. Multiplying these polynomials helps in analyzing the system's behavior and designing controllers. Similarly, in computer graphics, polynomial multiplication is used in curve and surface modeling.
