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Zorluk: KolayFactoring Polynomials

Which of the following represents the completely factored form of the algebraic expression 2x2−4x−302x^2 - 4x - 30?

  1. A
    2(x+5)(x−3)2(x + 5)(x - 3)
  2. B
    2(x2−2x−15)2(x^2 - 2x - 15)
  3. 2(x−5)(x+3)2(x - 5)(x + 3)Cevap
  4. D
    2(x−5)(x−3)2(x - 5)(x - 3)

Cevap

The completely factored form is 2(x−5)(x+3)2(x - 5)(x + 3).
To find the completely factored form, first factor out the greatest common factor (GCF), which is 22. This gives 2(x2−2x−15)2(x^2 - 2x - 15). Next, factor the trinomial inside the parentheses. You need two numbers that multiply to −15-15 and add up to −2-2. These numbers are −5-5 and 33. Substituting these into the binomial factors yields the final completely factored form: 2(x−5)(x+3)2(x - 5)(x + 3).

Adım Adım Çözüm

1
Factor out the greatest common factor (GCF) from all terms in the expression 2x2−4x−302x^2 - 4x - 30.
The GCF is 22. Factoring it out leaves 2(x2−2x−15)2(x^2 - 2x - 15).
Identifying and extracting the GCF is always the first step in simplifying and completely factoring a polynomial.
2
Factor the remaining quadratic trinomial x2−2x−15x^2 - 2x - 15 by finding two integers that multiply to −15-15 and add to −2-2.
The two integers are −5-5 and 33.
For a trinomial in the form x2+bx+cx^2 + bx + c, the constants in the binomial factors must multiply to cc and add to bb.
3
Write the expression as the product of the GCF and the two binomial factors.
The fully factored expression is 2(x−5)(x+3)2(x - 5)(x + 3).
This combines all factored components into their most simplified, prime form.

Anahtar Kavram

Factoring out a greatest common factor (GCF) followed by factoring a quadratic trinomial.
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Konular

Factoring Polynomials

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