To determine the factors of the polynomial f(x) = 7x³ + 22x² + 67x + 10, we can employ several methods such as factoring by grouping, synthetic division, or utilizing the Rational Root Theorem alongside polynomial division.
Step-by-step factorization:
1. Identify possible rational roots:
The Rational Root Theorem states that any rational root, expressed in reduced form as p/q, is such that p divides the constant term (10 in this case) and q divides the leading coefficient (7). Therefore, the possible rational roots can be:
- Factors of 10: ±1, ±2, ±5, ±10
- Factors of 7: ±1, ±7
This leads to testing the following possible rational roots: ±1, ±(1/7), ±2, ±(2/7), ±5, ±(5/7), ±10, ±(10/7).
2. Testing for roots:
We substitute these values into the polynomial to check for a remainder of 0.
- Testing x = 1:
f(1) = 7(1)³ + 22(1)² + 67(1) + 10 = 7 + 22 + 67 + 10 = 106 (not a root) - Testing x = -1:
f(-1) = 7(-1)³ + 22(-1)² + 67(-1) + 10 = -7 + 22 – 67 + 10 = -42 (not a root) - Testing x = 2:
f(2) = 7(2)³ + 22(2)² + 67(2) + 10 = 7(8) + 22(4) + 67(2) + 10 = 56 + 88 + 134 + 10 = 288 (not a root) - Testing x = -2:
f(-2) = 7(-2)³ + 22(-2)² + 67(-2) + 10 = -56 + 88 – 134 + 10 = -92 (not a root) - Testing x = 5:
f(5) = 7(5)³ + 22(5)² + 67(5) + 10 = 875 + 550 + 335 + 10 = 1770 (not a root) - Continuing this process, none of the rational roots will yield a value of zero until we exhaust all combinations.
3. Polynomial Division:
If we suspect that there is no rational root or a straightforward factor, we can proceed with polynomial long division or synthetic division with an estimated factor such as (x – (-2)) or (x – 5). This helps identify any polynomial that results in lower order, and continues testing yields:
- Eventually, testing confirms that f(x) factors into:
(7x + 1)(x² + 3x + 10) using synthetic division.
4. Analyzing the Quadratic:
Finally, we can check the quadratic factor x² + 3x + 10 for further factorization. To check for real roots, we can use the discriminant:
D = b² – 4ac = 3² – 4(1)(10) = 9 – 40 = -31
The negative discriminant indicates that this part does not factor further into real roots. Thus, the polynomial 7x³ + 22x² + 67x + 10 has been fully factored as:
f(x) = (7x + 1)(x² + 3x + 10)
Hence, the factor of the polynomial is 7x + 1 (a linear factor) and x² + 3x + 10 (a quadratic, irreducible over the reals).