Divide the following: $\frac{2 x^2-3 x-20}{x-4}$
Since the denominator is a quadrinomial and linear, we will use synthetic division to solve this problem.
First we must find the DIVISOR TERM by setting the divisor $=0$ and solving for $x$.
$0=$
$x=$
Answer (1)
Set the divisor x − 4 to zero and solve for x , obtaining x = 4 .
Use synthetic division to divide the polynomial 2 x 2 − 3 x − 20 by x − 4 .
Perform synthetic division with the divisor term 4 and the coefficients 2 , − 3 , − 20 .
The result of the synthetic division is the quotient 2 x + 5 and the remainder 0 , so the final answer is 2 x + 5 .
Explanation
Understanding the Problem We are asked to divide the polynomial 2 x 2 − 3 x − 20 by the linear term x − 4 . Since the divisor is a linear term, we can use synthetic division to solve this problem.
Finding the Divisor Term First, we need to find the value of x that makes the divisor equal to zero. So, we set x − 4 = 0 and solve for x . Adding 4 to both sides of the equation, we get x = 4 . This is the value we will use in the synthetic division.
Setting up Synthetic Division Now, we set up the synthetic division. We write down the coefficients of the polynomial 2 x 2 − 3 x − 20 , which are 2, -3, and -20. We also write down the divisor term, which is 4.
Performing Synthetic Division We perform the synthetic division as follows:
4 | 2 -3 -20
| 8 20
------------------
2 5 0
We bring down the first coefficient, which is 2. Then, we multiply 4 by 2 to get 8, and we write 8 under -3. We add -3 and 8 to get 5. Then, we multiply 4 by 5 to get 20, and we write 20 under -20. We add -20 and 20 to get 0.
Writing the Quotient and Remainder The result of the synthetic division gives us the coefficients of the quotient and the remainder. The last number in the bottom row is the remainder, which is 0. The other numbers in the bottom row are the coefficients of the quotient. Since the original polynomial was of degree 2 and we divided by a polynomial of degree 1, the quotient will be of degree 1. Therefore, the quotient is 2 x + 5 .
Final Answer Since the remainder is 0, the division is exact, and we have
x − 4 2 x 2 − 3 x − 20 = 2 x + 5.
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, engineers use polynomial division to analyze and design systems, such as control systems and signal processing systems. In computer graphics, polynomial division is used to perform operations on curves and surfaces. Moreover, polynomial division is a crucial tool in calculus for finding limits and derivatives of rational functions. Understanding polynomial division helps in simplifying complex expressions and solving equations in these real-world applications.
