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Difficulty: MediumFactoring Polynomials

To determine the critical points of a trajectory, a physicist needs to fully factor the quadratic expression 4x2−8x−604x^2 - 8x - 60. What is the completely factored equivalent of this expression?

  1. 4(x−5)(x+3)4(x - 5)(x + 3)Answer
  2. B
    4(x+5)(x−3)4(x + 5)(x - 3)
  3. C
    (4x−20)(x+3)(4x - 20)(x + 3)
  4. D
    4(x−5)(x−3)4(x - 5)(x - 3)

Answer

The completely factored form of the expression is 4(x−5)(x+3)4(x - 5)(x + 3).
The correct answer is found by first extracting the Greatest Common Factor (GCF), which is 44, leaving 4(x2−2x−15)4(x^2 - 2x - 15). Then, the inner trinomial x2−2x−15x^2 - 2x - 15 is factored into (x−5)(x+3)(x - 5)(x + 3) because −5-5 and 33 multiply to −15-15 and add to −2-2. Combining these gives the fully factored form 4(x−5)(x+3)4(x - 5)(x + 3).

Step-by-Step Solution

1
Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression 4x2−8x−604x^2 - 8x - 60.
The GCF of 44, −8-8, and −60-60 is 44. Factoring it out yields 4(x2−2x−15)4(x^2 - 2x - 15).
Extracting the GCF first simplifies the remaining polynomial, making the trinomial easier to factor.
2
Factor the remaining quadratic trinomial x2−2x−15x^2 - 2x - 15 by finding two numbers that multiply to the constant term (−15-15) and add to the middle term's coefficient (−2-2).
The numbers are −5-5 and 33, so the trinomial factors to (x−5)(x+3)(x - 5)(x + 3).
Expanding (x−5)(x+3)(x - 5)(x + 3) gives x2+3x−5x−15x^2 + 3x - 5x - 15, which simplifies back to x2−2x−15x^2 - 2x - 15.
3
Combine the GCF and the factored binomials to write the completely factored form.
4(x−5)(x+3)4(x - 5)(x + 3)
This represents the original expression broken down into its prime polynomial factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.
Estimated Time:1m 0s
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Topics

Factoring Polynomials

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