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Difficulty: HardFactoring Polynomials

A rectangular prism has a volume represented by the expression 3x3−5x2−12x+203x^3 - 5x^2 - 12x + 20 cubic centimeters. If the height of the prism is x−2x - 2 centimeters, which of the following expressions represents a possible length of the base of the prism, in centimeters, assuming the length and width are linear binomials with integer coefficients?

  1. A
    x−2x - 2
  2. B
    3x2−53x^2 - 5
  3. 3x−53x - 5Answer
  4. D
    3x−103x - 10
  5. E
    x+4x + 4

Answer

The correct answer is the expression 3x−53x - 5.
The polynomial representing the volume can be factored by grouping: 3x3−5x2−12x+20=x2(3x−5)−4(3x−5)=(3x−5)(x2−4)3x^3 - 5x^2 - 12x + 20 = x^2(3x - 5) - 4(3x - 5) = (3x - 5)(x^2 - 4). Factoring the difference of squares yields (3x−5)(x−2)(x+2)(3x - 5)(x - 2)(x + 2). Since the height is x−2x - 2, the remaining dimensions of the base must be 3x−53x - 5 and x+2x + 2. Therefore, the expression 3x−53x - 5 is a possible length of the base.

Step-by-Step Solution

1
Group the terms of the cubic polynomial representing the volume: 3x3−5x2−12x+203x^3 - 5x^2 - 12x + 20.
(3x3−5x2)−(12x−20)(3x^3 - 5x^2) - (12x - 20)
Grouping terms allows us to factor the polynomial by grouping.
2
Factor out the greatest common factor (GCF) from each group.
x2(3x−5)−4(3x−5)x^2(3x - 5) - 4(3x - 5)
The GCF of 3x33x^3 and 5x25x^2 is x2x^2, and the GCF of 12x12x and 2020 is 44.
3
Factor out the common binomial factor (3x−5)(3x - 5).
(3x−5)(x2−4)(3x - 5)(x^2 - 4)
This rewrites the polynomial as a product of a linear binomial and a quadratic binomial.
4
Factor the quadratic term x2−4x^2 - 4 using the difference of squares identity.
(3x−5)(x−2)(x+2)(3x - 5)(x - 2)(x + 2)
The expression x2−4x^2 - 4 is a difference of squares, which factors into (x−2)(x+2)(x - 2)(x + 2).
5
Divide the factored volume by the height, x−2x - 2, to find the possible dimensions of the base.
The possible dimensions for the length and width of the base are 3x−53x - 5 and x+2x + 2.
Volume is the product of length, width, and height. Since the height is x−2x - 2, the remaining factors represent the length and width.

Key Concept

Factoring polynomials by grouping and difference of squares.
Estimated Time:2m 0s
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Topics

Factoring Polynomials

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