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Zorluk: OrtaFactoring Polynomials

Which of the following represents the completely factored form of the expression 4x4−16x24x^4 - 16x^2?

  1. A
    4x2(x−2)24x^2(x - 2)^2
  2. B
    4x3(x−4)4x^3(x - 4)
  3. 4x2(x−2)(x+2)4x^2(x - 2)(x + 2)Cevap
  4. D
    (2x2−4x)2(2x^2 - 4x)^2
  5. E
    16x2(x2−1)16x^2(x^2 - 1)

Cevap

The completely factored form of the expression is 4x2(x−2)(x+2)4x^2(x - 2)(x + 2).
The correct answer is found by first identifying the greatest common factor of the terms in the polynomial 4x4−16x24x^4 - 16x^2, which is 4x24x^2. Factoring this out results in 4x2(x2−4)4x^2(x^2 - 4). Next, the binomial x2−4x^2 - 4 is recognized as a difference of squares and is factored into (x−2)(x+2)(x - 2)(x + 2). Putting these together gives the completely factored form 4x2(x−2)(x+2)4x^2(x - 2)(x + 2).

Adım Adım Çözüm

1
Identify and factor out the greatest common factor (GCF) of the terms in the expression 4x4−16x24x^4 - 16x^2.
The GCF of 4x44x^4 and 16x216x^2 is 4x24x^2. Factoring it out yields 4x2(x2−4)4x^2(x^2 - 4).
Factoring out the greatest common factor simplifies the polynomial and reveals remaining factorable patterns.
2
Factor the remaining binomial expression inside the parentheses, x2−4x^2 - 4.
Since x2−4x^2 - 4 is a difference of squares (x2−22x^2 - 2^2), it factors into (x−2)(x+2)(x - 2)(x + 2).
A difference of squares of the form a2−b2a^2 - b^2 always factors into (a−b)(a+b)(a - b)(a + b).
3
Combine the factored parts to write the completely factored expression.
4x2(x−2)(x+2)4x^2(x - 2)(x + 2)
This combines the GCF and the factored difference of squares to represent the original expression in its simplest factored parts.

Anahtar Kavram

Factoring a polynomial completely by first extracting the greatest common factor (GCF) and then applying the difference of squares formula.
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Konular

Factoring Polynomials

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