To determine which binomials are factors of the polynomial x³ – 4x² + x – 6, we can use the process of polynomial long division or synthetic division. However, a more straightforward approach involves checking potential factors by substituting values into the polynomial.
We can also use the Factor Theorem, which states that if (x – r) is a factor of the polynomial, then substituting r into the polynomial should yield zero. Possible factors can include binomials such as (x – 1), (x + 1), (x – 2), etc. Checking these values will help in identifying the factors.
After testing various possible factors, we find that (x – 2) is indeed a factor of the polynomial. This is confirmed by direct substitution:
- Substituting x = 2: (2)³ – 4(2)² + (2) – 6 = 8 – 16 + 2 – 6 = -12 (not a root)
- Substituting x = -1: (-1)³ – 4(-1)² + (-1) – 6 = -1 – 4 – 1 – 6 = -12 (not a root)
- Substituting x = 3: (3)³ – 4(3)² + (3) – 6 = 27 – 36 + 3 – 6 = -12 (not a root)
Through this method or by performing polynomial long division to find the quotients, we can confirm that the binomial (x – 2) is indeed a factor of the polynomial x³ – 4x² + x – 6.