Question

On a test, Leo is asked to completely factor the polynomial 3x3 – 3x + 5x2 – 5. He uses double grouping to get (x2 – 1)(3x + 5). Has he factored the polynomial completely? Explain.

Answer 1

Answer:

Leo does not  factored the polynomial completely.

3x^3-3x+5x^2-5=(x+1)(x-1)(3x+5)

Step-by-step explanation:

Given: On a test, Leo is asked to completely factor the polynomial

3x^3-3x+5x^2-5.

also, he uses double grouping to get (x^2- 1)(3x+5)

We have to explain whether he  factored the polynomial completely or not.

Consider the given polynomial 3x^3-3x+5x^2-5.

after double grouping he get (x^2- 1)(3x+5)

Also, we can simplify further ,

Using algebraic identity a^2-b^2=(a+b)(a-b)

We get,

(x^2- 1)=(x+1)(x-1)

We get, (x^2- 1)(3x+5)=(x+1)(x-1)(3x+5)

Thus, 3x^3-3x+5x^2-5=(x+1)(x-1)(3x+5)

Thus, He does not  factored the polynomial completely.

Answer 2

Answer:

No, Leo's answer is not a product of prime polynomials because x2 – 1 can be factored. This is a difference of squares. He should continue factoring to get

(x – 1)(x + 1)(3x + 5).

Step-by-step explanation: