To find the completely factored form of the polynomial f(x) = 6x³ + 13x² + 4x + 15, we can use the method of factoring by grouping or synthetic division, if necessary.
First, we can attempt to factor by grouping. We group the first two terms and the last two terms:
1. Grouping: (6x³ + 13x²) + (4x + 15)
2. Factoring out the greatest common factor from each group:
From 6x³ + 13x², the GCF is x²:
x²(6x + 13)
From 4x + 15, there is no common term, so we can leave it as (4x + 15).
Now, we rewrite the equation:
f(x) = x²(6x + 13) + (4x + 15)
Next, we still do not have a direct factoring solution, so let’s check for any rational roots using the Rational Root Theorem. We can try values such as ±1, ±3, ±5, ±15/(6), etc.
Upon testing, we can find that x = -3 is a root. This implies that (x + 3) is a factor.
Now, we perform synthetic division to divide the polynomial by (x + 3):
- Writing 6, 13, 4, 15 for the coefficients yields a new polynomial after synthetic division, which results in:
2x² + 7x + 5
Next, we need to factor 2x² + 7x + 5. We look for two numbers that multiply to 2 * 5 = 10 and add to 7, which are 5 and 2.
We rewrite:
2x² + 5x + 2x + 5 = 0
Factoring this gives:
(2x + 5)(x + 1)
Putting it all together, we have:
Final Factored Form:
f(x) = (x + 3)(2x + 5)(x + 1)
This is the completely factored form of the original polynomial.