A rectangular prism has a volume of [tex]4x^3 + 30x^2 + 36x[/tex]. What linear expressions can represent possible dimensions of the prism?

4 x 3 + 30 x 2 + 36 x = 2 x ( 2 x 2 + 15 x + 18 ) = 2 x ( 2 x 2 + 3 x + 12 x + 18 ) = = 2 x [ x ( 2 x + 3 ) + 6 ( 2 x + 3 )] = 2 x ( 2 x + 3 ) ( x + 6 ) t h e p oss ib l e d im e n s i o n s o f t h e p r i s m : a = 2 x , b = 2 x + 3 , c = x + 6 an d x  = 0 , x  = − 2 3 ​ , x  = − 6

Answered by kate200468 | 2024-06-10

The student is dealing with the factorization of a cubic polynomial that represents the volume of a rectangular prism. To find possible dimensions of the prism, we need to factor the polynomial 4x³ + 30x² + 36x . Starting with the greatest common factor, we extract x and get x(4x² + 30x + 36) . The quadratic can be factored further, and we find that (4x² + 30x + 36) breaks down into (2x + 6)(2x + 6) or (2x + 6)² . Thus, the volume can be expressed as x * (2x + 6) * (2x + 6) suggesting that the dimensions of the prism can be x, 2x + 6, and 2x + 6.

Answered by qwcat | 2024-06-24

The expression for the volume of the rectangular prism can be factored to yield possible dimensions as follows: a = 2 x , b = x + 3 , and c = x + 3 . The dimensions depend on the value of x being greater than 0 and not equal to -3. Therefore, dimensions are valid only for x  = 0 and x  = − 3 .
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Answered by kate200468 | 2024-12-23