Divide the following: $\frac{2 x^2-3 x-20}{x-4}$
Answer (1)
Set up the synthetic division with the coefficients of the dividend and the root of the divisor.
Perform the synthetic division by bringing down the leading coefficient, multiplying, and adding.
Read the coefficients of the quotient and the remainder from the last row of the synthetic division.
The result of the division is 2 x + 5 .
Explanation
Understanding the Problem We are asked to divide the quadratic polynomial 2 x 2 − 3 x − 20 by the linear expression x − 4 . This can be done using polynomial long division or synthetic division. Since the divisor is of the form x − a , synthetic division is a straightforward method.
Setting up Synthetic Division To 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 identify the value of a from the divisor x − 4 , which is a = 4 .
Performing Synthetic Division Now, we perform the synthetic division:
2
-3
-20
4
Bring down the leading coefficient (2):
2
-3
-20
4
2
Multiply 4 by 2 and write the result (8) under -3:
2
-3
-20
4
8
2
Add -3 and 8 to get 5:
2
-3
-20
4
8
2
5
Multiply 4 by 5 and write the result (20) under -20:
2
-3
-20
4
8
20
2
5
Add -20 and 20 to get 0:
2
-3
-20
4
8
20
2
5
0
Interpreting the Result The last row of the synthetic division (2, 5, 0) gives us the coefficients of the quotient and the remainder. The quotient is 2 x + 5 and the remainder is 0.
Final Answer Therefore, x − 4 2 x 2 − 3 x − 20 = 2 x + 5 .
Examples
Polynomial division is a fundamental concept in algebra and has many practical applications. For example, engineers use polynomial division to analyze the stability of control systems. Imagine designing a cruise control system for a car. The transfer function of the system can be represented as a rational function, which is a ratio of two polynomials. By performing polynomial division, engineers can simplify the transfer function and determine the system's response to different inputs, ensuring the cruise control system is stable and performs as expected. This ensures a smooth and safe driving experience.
