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Zorluk: OrtaFactoring Polynomials

An algebra student is asked to simplify a polynomial expression by factoring it completely. The student is given the expression 2x6−32x22x^6 - 32x^2. Which of the following is the completely factored form of the expression?

  1. A
    2x2(x2−4)(x2+4)2x^2(x^2 - 4)(x^2 + 4)
  2. B
    2x2(x3−16)2x^2(x^3 - 16)
  3. 2x2(x−2)(x+2)(x2+4)2x^2(x - 2)(x + 2)(x^2 + 4)Cevap
  4. D
    32x2(x−2)(x+2)(x2+4)32x^2(x - 2)(x + 2)(x^2 + 4)
  5. E
    2x2(x−2)2(x2+4)2x^2(x - 2)^2(x^2 + 4)

Cevap

The completely factored form of the expression is 2x2(x−2)(x+2)(x2+4)2x^2(x - 2)(x + 2)(x^2 + 4).
The correct answer is 2x2(x−2)(x+2)(x2+4)2x^2(x - 2)(x + 2)(x^2 + 4). First, find the greatest common factor (GCF) of 2x62x^6 and 32x232x^2, which is 2x22x^2. Factoring this out gives 2x2(x4−16)2x^2(x^4 - 16). Next, recognize that x4−16x^4 - 16 is a difference of squares since x4=(x2)2x^4 = (x^2)^2 and 16=4216 = 4^2. This factors into (x2−4)(x2+4)(x^2 - 4)(x^2 + 4). Finally, the term x2−4x^2 - 4 is also a difference of squares and factors into (x−2)(x+2)(x - 2)(x + 2), while the sum of squares x2+4x^2 + 4 cannot be factored further over the real numbers. Thus, the completely factored form is 2x2(x−2)(x+2)(x2+4)2x^2(x - 2)(x + 2)(x^2 + 4).

Adım Adım Çözüm

1
Identify and factor out the greatest common factor (GCF) of the two terms in 2x6−32x22x^6 - 32x^2.
The GCF is 2x22x^2, so the expression becomes 2x2(x4−16)2x^2(x^4 - 16).
Factoring out the GCF simplifies the polynomial and reveals a difference of squares.
2
Recognize that x4−16x^4 - 16 is a difference of squares and factor it.
Since x4=(x2)2x^4 = (x^2)^2 and 16=4216 = 4^2, the expression factors as 2x2(x2−4)(x2+4)2x^2(x^2 - 4)(x^2 + 4).
The difference of squares formula a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b) applies directly to x4−16x^4 - 16.
3
Check if any remaining factors can be factored further, and factor them.
The term x2−4x^2 - 4 is another difference of squares (x2−22x^2 - 2^2), which factors into (x−2)(x+2)(x - 2)(x + 2). The term x2+4x^2 + 4 is a sum of squares and cannot be factored over the real numbers. The completely factored expression is 2x2(x−2)(x+2)(x2+4)2x^2(x - 2)(x + 2)(x^2 + 4).
To factor completely, all factorable terms must be broken down into their prime polynomial factors.

Anahtar Kavram

Factoring polynomials completely using the greatest common factor and the difference of squares.
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Konular

Factoring Polynomials

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