How do you factor this completely:
x^3 + ax + 3a + 3x^2
You are in luck with this one. Two pairs of terms have x + 3 as a common factor.
x^3 + ax + 3a + 3x^2
= x^2(x + 3)+ a(x + 3)
= (x^2 + a)(x + 3)
= (x + ia)(x - ia)(x + 3)
(i is the square root of -1)
Two of the three roots are complex, for real a.
X^3+ax+3a+3x^2
x^2+3x^2+ax+3a
x(x+3)+a(x+3)
(x+a)(x+3)
x= -a, x= -3
To factor the given expression completely, we can follow these steps:
Step 1: Rearrange the terms:
x^3 + 3x^2 + ax + 3a
Step 2: Group the terms in pairs:
(x^3 + 3x^2) + (ax + 3a)
Step 3: Look for the highest common factor (HCF) in each pair:
x^2(x + 3) + a(x + 3)
Step 4: Notice that we have a common binomial factor of (x + 3):
(x^2 + a)(x + 3)
Thus, the completely factored form of the expression is (x^2 + a)(x + 3).