Divide using synthetic division.
$(x^4+13 x^3-64 x^2-20 x+16) \div (x-4)$
Answer (1)
Set up the synthetic division table with the coefficients of the dividend and the value of c.
Bring down the first coefficient.
Multiply the value of c by the brought-down coefficient and write the result under the next coefficient. Add the two numbers.
Repeat the previous step until all coefficients have been used. The last number is the remainder, and the other numbers are the coefficients of the quotient.
The quotient is x 3 + 17 x 2 + 4 x − 4 .
Explanation
Understanding the Problem We are asked to divide the polynomial x 4 + 13 x 3 − 64 x 2 − 20 x + 16 by x − 4 using synthetic division. Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − c . In this case, c = 4 .
Setting up Synthetic Division First, write down the coefficients of the polynomial: 1, 13, -64, -20, 16. Then, write down the value of c , which is 4.
First Step Set up the synthetic division table as follows:
4 | 1 13 -64 -20 16
|
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1
Bring down the first coefficient (1).
Second Step Multiply 4 by 1 and write the result (4) under 13. Add 13 and 4 to get 17.
4 | 1 13 -64 -20 16
| 4
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1 17
Third Step Multiply 4 by 17 and write the result (68) under -64. Add -64 and 68 to get 4.
4 | 1 13 -64 -20 16
| 4 68
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1 17 4
Fourth Step Multiply 4 by 4 and write the result (16) under -20. Add -20 and 16 to get -4.
4 | 1 13 -64 -20 16
| 4 68 16
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1 17 4 -4
Fifth Step Multiply 4 by -4 and write the result (-16) under 16. Add 16 and -16 to get 0.
4 | 1 13 -64 -20 16
| 4 68 16 -16
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1 17 4 -4 0
Final Result The numbers 1, 17, 4, and -4 are the coefficients of the quotient, and 0 is the remainder. Therefore, the quotient is x 3 + 17 x 2 + 4 x − 4 .
Examples
Synthetic division is a useful tool in various engineering and scientific fields. For instance, when designing control systems, engineers often need to analyze the stability of a system, which involves finding the roots of a characteristic polynomial. Synthetic division can help simplify the polynomial, making it easier to find these roots and assess the system's stability. This ensures that the designed system operates reliably and predictably.
