Use synthetic division to decide whether x=1 is a solution of the equation:
[tex]$-5 x^3-9 x^2+x+13 = 0$[/tex]
Yes or No
Answer (2)
Using synthetic division with the polynomial − 5 x 3 − 9 x 2 + x + 13 , we found that when x = 1 , the remainder is 0 . This indicates that x = 1 is a root of the polynomial. Therefore, the answer is Yes.
;
Set up synthetic division with the coefficients of the polynomial and x = 1 .
Perform synthetic division to find the remainder.
The remainder is 0, which means the value of the polynomial at x = 1 is 0.
Therefore, the answer is Y es .
Explanation
Understanding the Problem We are given the polynomial − 5 x 3 − 9 x 2 + x + 13 and asked to determine if x = 1 is a root using synthetic division. In other words, we want to find the value of the polynomial when x = 1 .
Setting up Synthetic Division We can use synthetic division to evaluate the polynomial at x = 1 . We set up the synthetic division table with the coefficients of the polynomial: -5, -9, 1, and 13.
Performing Synthetic Division Now, we perform synthetic division:
Bring down the first coefficient: -5. Multiply -5 by 1: -5. Add -5 to -9: -14. Multiply -14 by 1: -14. Add -14 to 1: -13. Multiply -13 by 1: -13. Add -13 to 13: 0.
The synthetic division is as follows:
1 | -5 -9 1 13
| -5 -14 -13
----------------
-5 -14 -13 0
The remainder is 0.
Interpreting the Result Since the remainder is 0, the value of the polynomial at x = 1 is 0. Therefore, x = 1 is a root of the polynomial.
Final Answer Since the value of the polynomial at x = 1 is 0, the answer is Yes.
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 analyze systems modeled by polynomials, such as control systems or signal processing filters. For example, if you have a system described by a polynomial and you want to check if a particular input value makes the system output zero, you can use synthetic division to quickly evaluate the polynomial at that input value. This can help determine stability or resonance conditions in the system.
