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If the volume of a cuboid is 3x2−27, then its possible dimensions are:
3,x2,−27x
3,3,3
3,x−3,x+3
3,x2,27x
3,x−3,x+3
Factor out the common term 3, then use the difference of squares identity (a² - b²) = (a - b)(a + b) to express the dimensions as product of 3, (x - 3), and (x + 3).
The volume of a cuboid is given by the algebraic expression 3x² - 27.
V=l×b×h
Factor out the common term 3, then use the difference of squares identity (a² - b²) = (a - b)(a + b) to express the dimensions as product of 3, (x - 3), and (x + 3).
Students often stop after factoring out 3, getting 3(x² - 9), or forget to identify that a cuboid must have three distinct dimensions (length, breadth, and height).
Factorization
Start by identifying the common factor of the volume expression 3x2−27. We factor out 3 to get 3(x2−9).
3x2−27=3(x2−9)
Apply Identity
Use the difference of squares identity a2−b2=(a−b)(a+b) where a=x and b=3.
x2−32=(x−3)(x+3)
Result
Combining these, the expression is 3(x−3)(x+3). Since the volume of a cuboid is l×b×h, the dimensions are 3,(x−3), and (x+3).
V=3×(x−3)×(x+3)
C is correct because the expression 3x2−27 factors into 3(x−3)(x+3), representing the three dimensions of the cuboid.
This concept of factoring polynomials is vital in Geometry for finding missing dimensions and in Calculus for finding stationary points of volume functions.
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