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Difficulty: MediumFactoring Polynomials

The area of a rectangular board is given by the algebraic expression 4x2−8x−54x^2 - 8x - 5. Which of the following shows the dimensions of the board expressed as a product of two binomials?

  1. (2x+1)(2x−5)(2x + 1)(2x - 5)Answer
  2. B
    (2x−1)(2x+5)(2x - 1)(2x + 5)
  3. C
    (4x+1)(x−5)(4x + 1)(x - 5)
  4. D
    (2x−1)(2x−5)(2x - 1)(2x - 5)

Answer

The correct dimensions are given by the factored form (2x+1)(2x−5)(2x + 1)(2x - 5).
To factor 4x2−8x−54x^2 - 8x - 5, find two values that multiply to −20-20 (which is 4×−54 \times -5) and add to −8-8. These values are −10-10 and 22. Splitting the middle term gives 4x2−10x+2x−54x^2 - 10x + 2x - 5. Factoring by grouping gives 2x(2x−5)+1(2x−5)2x(2x - 5) + 1(2x - 5), which then factors out to (2x+1)(2x−5)(2x + 1)(2x - 5).

Step-by-Step Solution

1
Identify the coefficients for the quadratic expression ax2+bx+cax^2 + bx + c.
a=4a = 4, b=−8b = -8, and c=−5c = -5.
To factor a non-monic quadratic trinomial by grouping, we first need to determine the product of aa and cc.
2
Find two numbers that multiply to the product of aa and cc (−20-20) and add to bb (−8-8).
The two numbers are −10-10 and 22.
These numbers allow us to split the middle term (−8x-8x) into two parts that can be factored by grouping.
3
Rewrite the original expression by splitting the middle term.
4x2−10x+2x−54x^2 - 10x + 2x - 5
This sets up the expression for factoring by grouping.
4
Factor the greatest common factor (GCF) from the first two terms and the last two terms.
2x(2x−5)+1(2x−5)2x(2x - 5) + 1(2x - 5)
Extracting the GCF reveals a common binomial factor in both groups.
5
Factor out the common binomial factor.
(2x+1)(2x−5)(2x + 1)(2x - 5)
This provides the completely factored form of the original area expression.

Key Concept

Factoring Non-Monic Quadratic Trinomials
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Topics

Factoring Polynomials

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