Use synthetic division to solve $(2 x^3+4 x^2-35 x+15) \div (x-3)$. What is the quotient?
A. $2 x^2-2 x-29+\frac{102}{x+3}$
B. $2 x^2-2 x-29+\frac{102}{x-3}$
C. $2 x^3+10 x^2-5 x$
D. $2 x^2+10 x-5$
Answer (1)
Set up the synthetic division with the root of the divisor (3) and the coefficients of the polynomial (2, 4, -35, 15).
Perform the synthetic division process.
Identify the coefficients of the quotient from the result of the synthetic division.
Write the quotient as 2 x 2 + 10 x − 5 .
Explanation
Understanding the Problem We are given the polynomial 2 x 3 + 4 x 2 − 35 x + 15 and asked to divide it by ( x − 3 ) using synthetic division. Our goal is to find the quotient.
Setting up Synthetic Division We set up the synthetic division with the root of the divisor, which is 3, and the coefficients of the polynomial: 2, 4, -35, and 15.
Performing Synthetic Division We perform synthetic division as follows:
Bring down the first coefficient (2).
Multiply it by 3: 2 × 3 = 6 .
Add the result to the next coefficient (4): 4 + 6 = 10 .
Multiply 10 by 3: 10 × 3 = 30 .
Add the result to the next coefficient (-35): − 35 + 30 = − 5 .
Multiply -5 by 3: − 5 × 3 = − 15 .
Add the result to the last coefficient (15): 15 + ( − 15 ) = 0 .
The numbers we obtained are 2, 10, -5, and 0.
Identifying Quotient and Remainder The last number obtained (0) is the remainder, and the other numbers (2, 10, -5) are the coefficients of the quotient. Since we started with a cubic polynomial (degree 3) and divided by a linear term (degree 1), the quotient will be a quadratic polynomial (degree 2).
Writing the Quotient Therefore, the quotient is 2 x 2 + 10 x − 5 , and the remainder is 0.
Final Answer The quotient of the division x − 3 2 x 3 + 4 x 2 − 35 x + 15 is 2 x 2 + 10 x − 5 .
Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . It's often used in engineering to simplify complex polynomial expressions that arise in control systems or signal processing. For instance, if you have a system described by a high-degree polynomial and you want to analyze its behavior around a specific point, synthetic division can quickly reduce the polynomial to a simpler form, making analysis and design easier. It also helps in finding roots of polynomials, which is crucial in stability analysis of systems.
