Final answer:
The synthetic division represents a cubic polynomial 2x^3 + 10x^2 + x + 5 being divided by x + 5, resulting in a remainder of 0 and a quotient of 2x^2 + 1.
Step-by-step explanation:
The synthetic division presented seems to represent the process of dividing a cubic polynomial by a linear factor. When we perform synthetic division, we are interested in dividing a polynomial by a binomial of the form (x - c). The coefficients of the polynomial are listed on the top row, and the constant from the linear divisor (x - c) is placed to the left of the vertical bar. In this case, -5 is our constant c, and the polynomial being divided is 2x^3 + 10x^2 + x + 5.
We then proceed to bring down the leading coefficient and multiply and add sequentially as per the rules of synthetic division. The result gives us a new set of coefficients, which represent the quotient polynomial, and the final value is the remainder of the division. Here, it seems like the remainder is 0, indicating that (x + 5) is a factor of the cubic polynomial.
The quotient in the given example will have coefficients 2, 0, 1, which corresponds to 2x^2 + 0x + 1 or simply 2x^2 + 1 as the dividend represented by the synthetic division is the original polynomial minus its remainder when divided by x + 5. The process of synthetic division is generally taught in algebra courses and is crucial for understanding polynomial division.