Final answer:
To find a factor of the polynomial 2x³-x²-2x+3, one would typically check for integer roots that would correspond to factors, commonly starting with factors of the constant term. In this case, the integers -1, 1, -3, and 3 are not roots of the polynomial, suggesting that if a rational root exists, it is a non-integer or a more complex algebraic factor is required to be found using advanced techniques.
Step-by-step explanation:
To find a factor of the polynomial 2x³-x²-2x+3, we can attempt to use the rational root theorem or synthetic division. However, without specific techniques mentioned, we can check for possible factors by plugging in integer values.
One common factor-checking method is to use synthetic division or the remainder theorem with possible roots derived from the factors of the constant term, which, for this polynomial, are ±1, ±3. If we find a value of x for which the polynomial equals zero, that value is a root, and x minus that value is a factor.
If we test x=1, we find that 2(1)³-(1)²-2(1)+3 does not equal zero. However, testing x=-1 gives 2(-1)³-(-1)²-2(-1)+3 = -2-1+2+3 = 2, which is not zero either. Next, if we try x=3, we get 2(3)³-(3)²-2(3)+3 = 54-9-6+3 = 42, which is not zero. Thus, -1 and 3 are not factors.
If we try x=-3, we find that 2(-3)³-(-3)²-2(-3)+3 = -54-9+6+3 = -54, which again isn't zero. So, no integer factors between -3 and 3 seem to work.
Without a clear factor evident among simple integer values, further methods such as the use of graphing calculators, factoring by grouping, or algebraic techniques like the Factor Theorem or synthetic division may be necessary to find a non-obvious factor.