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Zorluk: OrtaFactoring Polynomials

Which of the following is the completely factored form of the expression 3x3−12x2+12x3x^3 - 12x^2 + 12x?

  1. A
    3(x3−4x2+4x)3(x^3 - 4x^2 + 4x)
  2. B
    3x2(x−2)23x^2(x - 2)^2
  3. C
    3x(x−2)(x+2)3x(x - 2)(x + 2)
  4. 3x(x−2)23x(x - 2)^2Cevap
  5. E
    3x(x+2)23x(x + 2)^2

Cevap

The completely factored form of the expression is 3x(x−2)23x(x - 2)^2.
To factor the expression 3x3−12x2+12x3x^3 - 12x^2 + 12x completely, we first look for the greatest common factor (GCF) of the three terms. The GCF of 3x33x^3, −12x2-12x^2, and 12x12x is 3x3x. Factoring out 3x3x yields 3x(x2−4x+4)3x(x^2 - 4x + 4). Next, we factor the quadratic trinomial inside the parentheses. The expression x2−4x+4x^2 - 4x + 4 is a perfect square trinomial that factors into (x−2)2(x - 2)^2. Putting it all together, the completely factored form is 3x(x−2)23x(x - 2)^2.

Adım Adım Çözüm

1
Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial 3x3−12x2+12x3x^3 - 12x^2 + 12x.
3x(x2−4x+4)3x(x^2 - 4x + 4)
Each term in the polynomial is divisible by 33, and the lowest power of xx common to all terms is x1x^1. Thus, the GCF is 3x3x.
2
Factor the remaining quadratic trinomial x2−4x+4x^2 - 4x + 4 inside the parentheses.
(x−2)2(x - 2)^2
The trinomial x2−4x+4x^2 - 4x + 4 is a perfect square trinomial matching the form a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2, where a=xa = x and b=2b = 2.
3
Combine the factored parts to write the final completely factored expression.
3x(x−2)23x(x - 2)^2
Bringing together the GCF and the factored trinomial yields the simplest, fully factored form.

Anahtar Kavram

Factoring a polynomial completely by first extracting the greatest common factor (GCF) and then factoring the remaining perfect square trinomial.

Alternatif Yöntem

Instead of factoring directly, you can expand the answer choices to see which one is equivalent to the original polynomial. For example, expanding the correct option: 3x(x−2)2=3x(x2−4x+4)=3x3−12x2+12x3x(x-2)^2 = 3x(x^2 - 4x + 4) = 3x^3 - 12x^2 + 12x. This matches the original expression.
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Factoring Polynomials

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