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Question

Difficulty: MediumFactoring Polynomials

The area of a rectangle is represented by the expression 18x3−50x18x^3 - 50x square inches. If the width of the rectangle is 2x2x inches, which of the following expressions represents the length of the rectangle, in inches?

  1. A
    (3x−5)2(3x - 5)^2
  2. B
    9x−259x - 25
  3. C
    9x2−50x9x^2 - 50x
  4. (3x−5)(3x+5)(3x - 5)(3x + 5)Answer
  5. E
    18x3−52x18x^3 - 52x

Answer

The length of the rectangle is represented by the expression (3x−5)(3x+5)(3x - 5)(3x + 5).
The correct answer is the expression (3x−5)(3x+5)(3x - 5)(3x + 5) because the length is found by dividing the area, 18x3−50x18x^3 - 50x, by the width, 2x2x. Simplifying this quotient yields 9x2−259x^2 - 25. The expression 9x2−259x^2 - 25 is a difference of squares and factors completely into (3x−5)(3x+5)(3x - 5)(3x + 5).

Step-by-Step Solution

1
Set up the equation for the length of the rectangle by dividing the area by the width.
Length=18x3−50x2x\text{Length} = \frac{18x^3 - 50x}{2x}
Since the area of a rectangle is the product of its length and width, the length can be found by dividing the area by the width.
2
Perform the polynomial division by dividing each term of the numerator by the denominator 2x2x.
18x3−50x2x=18x32x−50x2x=9x2−25\frac{18x^3 - 50x}{2x} = \frac{18x^3}{2x} - \frac{50x}{2x} = 9x^2 - 25
Dividing a polynomial by a monomial requires dividing every term of the polynomial by that monomial.
3
Factor the resulting expression 9x2−259x^2 - 25 using the difference of squares identity, a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
9x2−25=(3x)2−52=(3x−5)(3x+5)9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)
The quadratic expression is a difference of two perfect squares, which can be factored into a product of binomial conjugates.

Key Concept

Factoring a difference of squares after dividing a polynomial by a monomial
Estimated Time:1m 30s
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Topics

Factoring Polynomials

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