Question

Factor the given polynomial by finding the greatest common monomial factor (or the negative of the greatest common monomial factor) and rewrite the expression.

2x^2+2

Answer 1

To factor the given polynomial 2x^2 + 2 by finding the greatest common monomial factor, we need to find the largest monomial that evenly divides both terms.

Looking at the coefficients of each term, we see that both terms share a common factor of 2. However, since there is no x in the second term, x is not common to both terms and cannot be included in the greatest common factor.

So, the greatest common monomial factor for 2x^2 + 2 is just 2. We then divide each term by 2 to find the other factor:

2x^2 ÷ 2 = x^2
2 ÷ 2 = 1

Now, we rewrite the original expression with the greatest common factor factored out:

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

The expression inside the parentheses, x^2 + 1, cannot be factored further over the real numbers because there are no real number solutions that would make x^2 + 1 equal to zero (it's irreducible). Consequently, 2(x^2 + 1) is the factored form of the polynomial with the greatest common monomial factor extracted.