In Exercises 17–32, divide using synthetic division. (x⁵+4x⁴−3x²+2x+3)÷(x−3)

Verified step by step guidance

1. Identify the dividend and divisor. The dividend is the polynomial $x^5+\; 4x^4-\; 3x^2+\; 2x\; +\; 3$, and the divisor is $x\; -\; 3$.
2. Set up synthetic division by writing the coefficients of the dividend in descending order of powers of x. Since the polynomial is missing the $x^3$ term, include a 0 for its coefficient. The coefficients are: 1, 4, 0, -3, 2, 3.
3. Write the zero of the divisor $x\; -\; 3$, which is $3$, to the left side for synthetic division.
4. Perform synthetic division by bringing down the first coefficient, multiplying it by 3, adding to the next coefficient, and repeating this process across all coefficients.
5. Interpret the final row of numbers as the coefficients of the quotient polynomial, starting from one degree less than the dividend, and the last number as the remainder.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.

Polynomial Coefficients and Missing Terms

When performing synthetic division, it is important to include coefficients for all powers of x, even if some terms are missing. For example, if x^3 is missing, its coefficient is zero. This ensures the alignment of terms during the division process.

Interpreting the Result of Synthetic Division

The result of synthetic division gives the quotient polynomial and a remainder. The quotient has one degree less than the original polynomial, and the remainder is a constant. The original polynomial equals the divisor times the quotient plus the remainder.