Use synthetic division to solve $(2 x^3+4 x^2-35 x+15) \div (x-3)$. What is the quotient?
A. $2 x^2-2 x-28+\frac{102}{x+3}$
B. $2 x^2-2 x-29+\frac{m}{x-3}$
C. $2 x^3+10 x^2-5 x$
D. $2 x^2+10 x-5$

Set up the synthetic division with the coefficients of the polynomial and the root of the divisor.
Perform the synthetic division.
Identify the coefficients of the quotient from the result of the synthetic division.
Write the quotient polynomial: 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 we want to divide it by ( x − 3 ) using synthetic division. Our goal is to find the quotient.

Setting up Synthetic Division We will set up the synthetic division table with the coefficients of the polynomial 2 x 3 + 4 x 2 − 35 x + 15 and the root of the divisor ( x − 3 ) , which is 3 . The coefficients are 2 , 4 , − 35 , 15 .

Performing Synthetic Division Now, we perform synthetic division:
3 | 2 4 -35 15



6 30 -15



2 10 -5 0



The numbers in the bottom row are the coefficients of the quotient and the remainder.

Determining the Quotient The resulting coefficients from the synthetic division are 2 , 10 , − 5 , and 0 . These represent the coefficients of the quotient polynomial and the remainder. 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). The remainder is 0. Therefore, the quotient is 2 x 2 + 10 x − 5 .

Final Answer The quotient of the division 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 is often used in algebra to find the roots of polynomials, simplify expressions, and solve equations. For example, if you are designing a bridge and need to calculate the bending moment of a beam, you might encounter a polynomial expression. Using synthetic division can help simplify this expression, making it easier to analyze the structural integrity of the bridge. Synthetic division helps engineers and scientists simplify complex calculations, making it a valuable tool in various fields.

Answered by GinnyAnswer | 2025-07-05

Using synthetic division, we find that the quotient of ( 2 x 3 + 4 x 2 − 35 x + 15 ) ÷ ( x − 3 ) is 2 x 2 + 10 x − 5 . Therefore, the answer is option D. The division resulted in no remainder, confirming the completeness of our solution.
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Answered by Anonymous | 2025-07-08