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Zorluk: KolayFactoring Polynomials

Consider the algebraic expression: 3x2+6x−243x^2 + 6x - 24. When this expression is factored completely, which of the following represents the correct product?

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

Cevap

The completely factored form is 3(x+4)(x−2)3(x + 4)(x - 2).
To factor the expression completely, you must first factor out the greatest common factor (GCF) of 3 from all terms. This leaves 3(x2+2x−8)3(x^2 + 2x - 8). Next, you factor the quadratic trinomial inside the parentheses. You need two numbers that multiply to -8 and add to +2. Those numbers are +4 and -2. Therefore, combining the GCF and the binomials gives the completely factored form: 3(x+4)(x−2)3(x + 4)(x - 2).

Adım Adım Çözüm

1
Identify and factor out the greatest common factor (GCF) from all three terms in the expression 3x2+6x−243x^2 + 6x - 24.
The GCF is 3. Factoring it out yields 3(x2+2x−8)3(x^2 + 2x - 8).
Factoring out the GCF simplifies the remaining trinomial, making it easier to factor and ensuring the final answer is factored completely.
2
Factor the remaining quadratic trinomial x2+2x−8x^2 + 2x - 8 by finding two integers that multiply to the constant term (-8) and add up to the middle coefficient (2).
The correct integers are 4 and -2 (since 4 * -2 = -8 and 4 + -2 = 2). This gives the binomial factors (x+4)(x−2)(x + 4)(x - 2).
This process decomposes the monic quadratic trinomial into its constituent binomial factors.
3
Combine the GCF and the binomial factors to write the final factored expression.
The final product is 3(x+4)(x−2)3(x + 4)(x - 2).
This represents the original expression as a product of its simplest prime factors.

Anahtar Kavram

Factoring Polynomials using GCF and Trinomial Decomposition
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Konular

Factoring Polynomials

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