Question

Which polynomial is factored completely?

121x^2+36y^2

(4x+4)(x+1)

2x(x^2-4)

3x^4-15n^3+12n^2

Answer 1

Upon review of the provided polynomials, none are factored completely. To be completely factored, a polynomial must be simplified into its simplest form without any further factors remaining. All the given options can be factored further or simplified.

Step-by-step explanation:

The question asks which polynomial is factored completely. Let's analyze each option:

1. 121x^2+36y^2 is a sum of squares and cannot be factored over the real numbers, so it is not completely factored.

2. (4x+4)(x+1) is already in factored form, but we can factor out a 4 from the first term to get 4(x+1)(x+1) or 4(x+1)^2, so the original form was not completely factored.

3. 2x(x^2-4) can be further factored using the difference of squares: 2x(x+2)(x-2), which shows that the original form was not completely factored.

4. 3x^4-15x^3+12x^2 can be factored by pulling out the common factor of x^2, giving us x^2(3x^2-15x+12). We can further factor this; the quadratic can be factored as 3(x^4-5x^3+4x^2), which can be simplified to 3x^2(x-1)(x-4). This indicates that it was not factored completely.

None of the given polynomials are completely factored. To be factored completely, all polynomials must not have any further factors and should be reduced to their simplest form with all common factors pulled out.

Answer 2

The polynomial 2x(x^2-4) is factored completely.

Step-by-step explanation:

In order to determine which polynomial is factored completely, we need to look at the given options:

1. 121x^2+36y^2 - This is in the form of a sum of squares and cannot be factored further.

2. (4x+4)(x+1) - This is the factored form of a quadratic polynomial and is already completely factored.

3. 2x(x^2-4) - This can be factored further by recognizing that the term (x^2-4) is the difference of squares, resulting in 2x(x-2)(x+2).

4. 3x^4-15n^3+12n^2 - This polynomial is not factored completely and cannot be simplified any further.

Given these options, the polynomial that is factored completely is 2x(x^2-4).