Final answer:
The likely error is a mistyped polynomial. Synthetic division of the corrected polynomial x^2 - 5x + 3 by x - 2 follows a step-by-step process resulting in a quotient of x - 3 with a remainder of -3.
Step-by-step explanation:
The student has made an error in typing their polynomial. It is likely they intended to write x^2 - 5x + 3 for division by x - 2 using synthetic division. Nonetheless, if we correct this as x^2 - 5x + 3, the process of synthetic division would start by using the zero of the divisor, which is +2 in this case, as the synthetic number.
- Write down the coefficients of the polynomial: 1 (from x^2), -5 (from -5x), and +3 (constant term).
- Bring down the leading coefficient (1).
- Multiply this leading coefficient by the synthetic number (2), and place the result (2) underneath the second coefficient (-5).
- Add the second column giving the new second coefficient (-5 + 2 = -3).
- Multiply the new second coefficient (-3) by the synthetic number (2), and place the result (-6) underneath the third coefficient (+3).
- Add the third column giving the remainder (-6 + 3 = -3).
The result of the synthetic division of x^2 - 5x + 3 by x - 2 is an equation of the first degree with a remainder, given as x - 3 with a remainder of -3.