In Exercises 17–32, divide using synthetic division. (x⁷+x⁵−10x³+12)/(x+2)

Verified step by step guidance

1. Identify the divisor and the dividend. Here, the divisor is $x+2$, and the dividend is $x^7+x^5-\; 10x^3+\; 12$.
2. Write down the coefficients of the dividend in descending order of powers of $x$. For missing powers, use 0 as the coefficient. The coefficients are: 1 (for $x^7$), 0 (for $x^6$), 1 (for $x^5$), 0 (for $x^4$), -10 (for $x^3$), 0 (for $x^2$), 0 (for $x^1$), and 12 (constant term).
3. Set up synthetic division by writing the opposite of the constant term in the divisor. Since the divisor is $x\; +\; 2$, use $-2$ as the synthetic divisor.
4. Perform synthetic division: bring down the first coefficient, multiply it by $-2$, add to the next coefficient, and repeat this process across all coefficients.
5. The final row of numbers after synthetic division will give the coefficients of the quotient polynomial, and the last number will be 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 and performing arithmetic operations in a tabular form. This method is efficient for finding quotients and remainders quickly.

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. Missing terms should be represented by zero coefficients to maintain the correct alignment during division. For example, x^7 + x^5 − 10x^3 + 12 should be written with zeros for x^6, x^4, x^2, and x terms.

Dividing by a Binomial of the Form x + c

Synthetic division requires the divisor to be in the form x - c. If the divisor is x + 2, rewrite it as x - (-2), so c = -2. This value is used in the synthetic division process to perform the calculations and find the quotient and remainder.