Use synthetic division to solve $(3 x^4+6 x^3+2 x^2+9 x+10) \div (x+2)$. What is the quotient?
Answer (1)
Set up the synthetic division with the root of the divisor (-2) and the coefficients of the polynomial (3, 6, 2, 9, 10).
Perform the synthetic division to obtain the coefficients of the quotient and the remainder.
The coefficients of the quotient are 3, 0, 2, and 5, and the remainder is 0.
The quotient is 3 x 3 + 2 x + 5 , so the final answer is 3 x 3 + 2 x + 5 .
Explanation
Understanding the Problem We are given the polynomial 3 x 4 + 6 x 3 + 2 x 2 + 9 x + 10 and asked to divide it by x + 2 using synthetic division. Our goal is to find the quotient.
Setting up Synthetic Division To perform synthetic division, we set up the division with the coefficients of the polynomial and the root of the divisor. The divisor is x + 2 , so the root is x = − 2 . The coefficients of the polynomial are 3, 6, 2, 9, and 10.
Performing Synthetic Division We perform synthetic division as follows:
-2 | 3 6 2 9 10 | -6 0 -4 -10 ---------------------- 3 0 2 5 0
The first number, 3, is brought down. Then, we multiply -2 by 3 to get -6, and add it to 6 to get 0. Next, we multiply -2 by 0 to get 0, and add it to 2 to get 2. Then, we multiply -2 by 2 to get -4, and add it to 9 to get 5. Finally, we multiply -2 by 5 to get -10, and add it to 10 to get 0.
Determining the Quotient The numbers 3, 0, 2, and 5 are the coefficients of the quotient, and the last number, 0, is the remainder. Therefore, the quotient is 3 x 3 + 0 x 2 + 2 x + 5 , which simplifies to 3 x 3 + 2 x + 5 .
Final Answer The quotient of the division is 3 x 3 + 2 x + 5 .
Examples
Polynomial division, like the one we just performed, is useful in many areas of mathematics and engineering. For example, engineers use polynomial division when designing filters for signal processing. Suppose an engineer wants to design a filter with a certain frequency response. The transfer function of the filter can often be expressed as a rational function, which is a ratio of two polynomials. To simplify the design, the engineer might need to divide the polynomials to reduce the complexity of the transfer function. This can help in creating a more efficient and cost-effective filter design. In general, polynomial division helps simplify complex expressions, making them easier to analyze and implement in real-world applications.
