To factor the polynomial [tex]\( x^3 + 1 \)[/tex] with integer coefficients, follow these steps:
1. Recognize the form of the polynomial:
The polynomial [tex]\( x^3 + 1 \)[/tex] is a sum of cubes. A sum of cubes can be factored using the formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
2. Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
In our polynomial [tex]\( x^3 + 1 \)[/tex], we can identify [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex].
3. Apply the sum of cubes formula:
Substitute [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex] into the formula:
[tex]\[
x^3 + 1^3 = (x + 1)(x^2 - x \cdot 1 + 1^2)
\][/tex]
4. Simplify the expression:
Perform the multiplications and additions inside the second factor:
[tex]\[
x^2 - x \cdot 1 + 1^2 = x^2 - x + 1
\][/tex]
5. Combine the factors:
Substitute the simplified second factor back:
[tex]\[
x^3 + 1 = (x + 1)(x^2 - x + 1)
\][/tex]
Thus, the factorization of [tex]\( x^3 + 1 \)[/tex] with integer coefficients is:
[tex]\[
x^3 + 1 = (x + 1)(x^2 - x + 1)
\][/tex]
This is the fully factored form of the polynomial with integer coefficients.
