Certainly! Let's determine the missing factor step-by-step.
Given the factor: [tex]\( x^2 + x + 1 \)[/tex].
This is a quadratic polynomial.
We know the polynomial in question is a 3rd degree polynomial. When multiplied by another polynomial, [tex]\( x^2 + x + 1 \)[/tex] should result in a polynomial of degree 3. Therefore, the other factor must be a linear polynomial to ensure their product results in a cubic polynomial.
Let's assume the other factor is a linear polynomial of the form [tex]\( x + A \)[/tex], where [tex]\( A \)[/tex] is a constant. So the given polynomial can be expressed as:
[tex]\[ (x^2 + x + 1)(x + A) \][/tex]
We now need to expand this and ensure the resulting polynomial matches the characteristics of a cubic polynomial.
[tex]\[
(x^2 + x + 1)(x + A) = x^3 + Ax^2 + x^2 + Ax + x + A
= x^3 + (A+1)x^2 + (A+1)x + A
\][/tex]
For this to be a polynomial consistent with a standard cubic form, the terms must properly align to ensure the degrees and coefficients match. We compare this standard form [tex]\( ax^3 + bx^2 + cx + d \)[/tex] to our expanded form [tex]\( x^3 + (A+1)x^2 + (A+1)x + A \)[/tex].
To satisfy the requirement that we get a polynomial of degree 3, the linear factor must essentially fill the gap without modifying the coefficient of the highest-degree term. Therefore, comparing the degrees and coefficients, the linear factor needs to be [tex]\( x + 0 \)[/tex] or just [tex]\( x \)[/tex].
Therefore, the missing factor is:
[tex]\[
\boxed{x}
\][/tex]
