To find all the factors of the polynomial function f(x) = x³ + 4x² + 20x + 48, given that one root is x = 6, we can apply the Remainder Theorem.
According to the Remainder Theorem, if c is a root of the polynomial f(x), then f(c) = 0. Since we know f(6) = 0, it confirms that (x – 6) is a factor of f(x).
The next step is to perform polynomial long division or synthetic division to divide f(x) by (x – 6).
Using synthetic division with 6:
- Write the coefficients: 1 (for x³), 4 (for x²), 20 (for x), and 48 (constant).
- Set up the synthetic division:
6
__________
1 4 20 48
\ 6
__________
Now, proceed with the division:
1
4
20
48
\
6
6 \
____
0
The result will give you a quotient of x² + 10 and a remainder of 0. This tells us:
f(x) = (x – 6)(x² + 10)
The next step is to factor x² + 10, which does not have real roots (the discriminant is negative). Therefore, we find that:
x² + 10 = (x – i√10)(x + i√10)
Finally, the complete factorization of f(x) is:
f(x) = (x – 6)(x – i√10)(x + i√10)
In conclusion, the factors of the polynomial function f(x) are:
- (x – 6)
- (x – i√10)
- (x + i√10)