Describe how to find linear expressions for the possible dimensions of a rectangular prism with a volume of
8k(cubed) + 26K(squared) +6K.
To find the possible dimensions of a rectangular prism with a given volume of 8k^3 + 26k^2 + 6k, we need to factorize the volume expression.
Step 1: Factorize the given volume expression
The volume expression is 8k^3 + 26k^2 + 6k. We can factorize it by finding the common factor, if any, among the terms. In this case, the common factor is k, so we can rewrite the expression as k(8k^2 + 26k + 6).
Step 2: Factorize the quadratic expression
The quadratic expression 8k^2 + 26k + 6 is a trinomial, so we need to factorize it further. To do this, we need to find two binomials that multiply to give this quadratic expression.
We can use the fact that given a quadratic expression in the form ax^2 + bx + c, the binomials will have the form (px + q)(rx + s), where p, q, r, and s are numbers.
In this case, we are looking for two binomials that multiply to give 8k^2 + 26k + 6. We can try different combinations until we find the correct one.
(2k + 1)(4k + 6) = 8k^2 + 26k + 6
So, we have successfully factored the quadratic expression as (2k + 1)(4k + 6).
Step 3: Write the linear expressions for possible dimensions
Now that we have factored the expression completely, we can write the linear expressions for the possible dimensions.
The possible dimensions of the rectangular prism can be represented by the terms in the factored expression: k, 2k + 1, and 4k + 6. Therefore, the linear expressions for the possible dimensions are:
Length: 2k + 1
Width: k
Height: 4k + 6
Remember that these linear expressions represent the possible dimensions of the rectangular prism for the given volume expression.