Understanding the Problem:
We start with the polynomial function f(x) = x³ + 4x² + 20x + 48. We know that one of its roots is x = 6. This means that when we substitute 6 into the polynomial, f(6) should equal 0.
Using the Remainder Theorem:
The Remainder Theorem states that if a polynomial f(x) is divided by x – r, the remainder of this division is f(r). Since we already know that f(6) = 0, we can conclude that x – 6 is a factor of f(x).
Performing Polynomial Long Division:
Next, we can perform polynomial long division to divide f(x) by x – 6. Here’s how it works step by step:
- Divide the leading term of x³ by the leading term of x to get x².
- Multiply the entire divisor (x – 6) by x² and subtract from f(x).
- This yields a new polynomial to simplify.
- Repeat the process until you cannot divide anymore.
Finding All Factors:
After performing the division, suppose you arrive at:
- Once the polynomial division is complete, let’s say we obtained x² + 10x + 8.
- Now, we need to factor x² + 10x + 8.
- This factors further to (x + 2)(x + 8).
Final Factors:
Now, we can compile all the factors of the original polynomial:
- f(x) = (x – 6)(x + 2)(x + 8)
Summary:
In conclusion, the factors of the polynomial function f(x) = x³ + 4x² + 20x + 48 are (x – 6), (x + 2), and (x + 8). Thank you for exploring polynomial factoring using the Remainder Theorem!