To determine if (x + 3) is a factor of a given polynomial, you can use the factoring and polynomial division methods as follows:
Factoring method:
Substitute -3 for x in the polynomial and evaluate it.
If the result is equal to zero, then (x + 3) is a factor of the polynomial.
For example, let's consider the polynomial P(x) = x^3 + 6x^2 + 11x + 6.
Substituting x = -3 in P(x), we get P(-3) = (-3)^3 + 6(-3)^2 + 11(-3) + 6 = -27 + 54 - 33 + 6 = 0.
Since the result is zero, we can conclude that (x + 3) is a factor of P(x).
Polynomial division method:
Divide the given polynomial by (x + 3) using polynomial long division or synthetic division.
If the remainder is zero, then (x + 3) is a factor of the polynomial.
For example, using the polynomial division method with P(x) = x^3 + 6x^2 + 11x + 6:
x^2 + 3x + 2
x + 3 | x^3 + 6x^2 + 11x + 6
x^3 + 3x^2
----------
3x^2 + 11x
3x^2 + 9x
----------
2x + 6
2x + 6
-----
0
The quotient is x^2 + 3x + 2, and the remainder is zero.
Therefore, (x + 3) is a factor of P(x).