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Zorluk: KolayFactoring Polynomials

A rectangular garden has an area given by the expression 4n2−4n−244n^2 - 4n - 24. Which of the following represents the completely factored form of this area expression?

  1. 4(n−3)(n+2)4(n - 3)(n + 2)Cevap
  2. B
    4(n+3)(n−2)4(n + 3)(n - 2)
  3. C
    (4n−12)(n+2)(4n - 12)(n + 2)
  4. D
    4(n−6)(n+1)4(n - 6)(n + 1)

Cevap

The completely factored form is 4(n−3)(n+2)4(n - 3)(n + 2).
To factor the expression completely, first remove the greatest common factor of 4 from all terms, which gives 4(n2−n−6)4(n^2 - n - 6). Then, factor the trinomial inside the parentheses. We need two numbers that multiply to -6 and add to -1, which are -3 and 2. This gives the completely factored form 4(n−3)(n+2)4(n - 3)(n + 2).

Adım Adım Çözüm

1
Identify and factor out the greatest common factor (GCF) from all terms in the polynomial 4n2−4n−244n^2 - 4n - 24.
4(n2−n−6)4(n^2 - n - 6)
Factoring out the GCF simplifies the polynomial and is the required first step for complete factorization.
2
Factor the remaining quadratic trinomial n2−n−6n^2 - n - 6 by finding two numbers that multiply to -6 and add to -1.
The numbers are -3 and 2, yielding (n−3)(n+2)(n - 3)(n + 2).
This breaks down the trinomial into its constituent binomial factors.
3
Combine the GCF and the binomial factors to write the final expression.
4(n−3)(n+2)4(n - 3)(n + 2)
This represents the full, completely factored form of the original area expression.

Anahtar Kavram

Factoring out a GCF followed by factoring a quadratic trinomial
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Konular

Factoring Polynomials

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