91Ó°ÊÓ

The Greatest Common Factor (GCF) is the largest number that can evenly divide all the terms in a polynomial. To find the GCF, you need to identify the common factors of each term.

In the expression given: 6x^2 + 6x + 6, let's break it down:
- The terms are 6x^2, 6x, and 6.
- The factors of 6x^2 are 1, 2, 3, 6, x, and x^2.
- The factors of 6x are 1, 2, 3, 6, and x.
- The factors of 6 are 1, 2, 3, and 6.

The highest number that appears in all lists is 6. Therefore, the GCF is 6. Factoring out the GCF helps simplify the polynomial and makes further factoring easier.

Quadratic expressions are polynomials of degree 2. They typically take the form of ax^2 + bx + c, where a, b, and c are constants. In our example, x^2 + x + 1 is a quadratic expression.

Every quadratic expression can be written in standard form and examined for factorization. Some quadratics can be factored into products of binomials. However, in cases where they cannot be factored easily, other methods such as completing the square or using the quadratic formula might be used.

For our quadratic expression x^2 + x + 1, we need to determine if it can be factored further. We'll use the discriminant, which is a handy tool for this purpose.

The discriminant of a quadratic expression, typically represented as \(\triangle\), helps determine the nature of the roots. It is calculated using the formula: \[ \triangle = b^2 - 4ac \]

For our quadratic expression, x^2 + x + 1, the coefficients are: \begin{align*} a &= 1, \ b &= 1, \ c &= 1 \end{align*}. Therefore, the discriminant is \(\triangle = 1^2 - 4(1)(1) = 1 - 4 = -3\). Since the discriminant is negative, this means that the quadratic expression has no real roots and cannot be factored over the real numbers.

In this case, since the discriminant is less than zero, we can conclude that the quadratic expression x^2 + x + 1 cannot be factored further. The final factorization results in: 6(x^2 + x + 1).