In Exercises 17–32, divide using synthetic division. (3x²+7x−20)÷(x+5)

Verified step by step guidance

1. Identify the divisor and set it equal to zero to find the value to use in synthetic division. For the divisor $x+5$, set $x+\; 5\; =\; 0$ which gives $x=\; -5$.
2. Write down the coefficients of the dividend polynomial $3x^2\; +\; 7x\; -\; 20$. The coefficients are $3$, $7$, and $-20$.
3. Set up the synthetic division by writing the value $-5$ on the left and the coefficients $3$, $7$, and $-20$ in a row to the right.
4. Perform synthetic division: bring down the first coefficient $3$, multiply it by $-5$, write the result under the next coefficient, add down the column, and repeat this process for all coefficients.
5. Interpret the final row of numbers: the first two numbers represent the coefficients of the quotient polynomial, and the last number is the remainder. Write the quotient in polynomial form with one degree less than the original dividend.

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 polynomials, making calculations faster and less error-prone.

Polynomial Coefficients

Polynomial coefficients are the numerical factors in front of the variable terms. In synthetic division, you work directly with these coefficients, arranging them in descending order of degree and including zeros for any missing terms.

Division by a Binomial of the Form x + c

When dividing by a binomial like x + 5, synthetic division uses the root of the divisor, which is -5 in this case. This value is used in the synthetic division process to find the quotient and remainder efficiently.