Use synthetic division to decide whether the given number is a solution of the given equation.

[tex]$-4 x^3+x^2-9 x-8 ; x=2$[/tex]

Yes
No

Using synthetic division on the polynomial − 4 x 3 + x 2 − 9 x − 8 with x = 2 , we found the remainder to be − 54 . Since the remainder is not zero, x = 2 is not a solution. Therefore, the answer is N o ​ .
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Answered by Anonymous | 2025-07-04

Perform synthetic division with x = 2 on the polynomial − 4 x 3 + x 2 − 9 x − 8 .
The remainder is − 54 .
Since the remainder is not 0, x = 2 is not a solution.
The answer is N o ​ .

Explanation

Understanding the Problem We are given the polynomial p ( x ) = − 4 x 3 + x 2 − 9 x − 8 and we want to determine if x = 2 is a solution of the equation p ( x ) = 0 using synthetic division. In other words, we want to check if p ( 2 ) = 0 .

Performing Synthetic Division We can use synthetic division to evaluate p ( 2 ) . We set up the synthetic division as follows:


2 | -4 1 -9 -8
|
----------------

Bring down the leading coefficient -4:
2 | -4 1 -9 -8
|
----------------
-4

Multiply 2 by -4 and write the result under 1:
2 | -4 1 -9 -8
| -8
----------------
-4

Add 1 and -8:
2 | -4 1 -9 -8
| -8
----------------
-4 -7

Multiply 2 by -7 and write the result under -9:
2 | -4 1 -9 -8
| -8 -14
----------------
-4 -7

Add -9 and -14:
2 | -4 1 -9 -8
| -8 -14
----------------
-4 -7 -23

Multiply 2 by -23 and write the result under -8:
2 | -4 1 -9 -8
| -8 -14 -46
----------------
-4 -7 -23

Add -8 and -46:
2 | -4 1 -9 -8
| -8 -14 -46
----------------
-4 -7 -23 -54

The remainder is -54.

Conclusion Since the remainder is -54, which is not equal to 0, x = 2 is not a solution of the equation − 4 x 3 + x 2 − 9 x − 8 = 0 .

Examples
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x − a . It's particularly useful in determining whether a given value 'a' is a root (or zero) of the polynomial. For example, engineers might use synthetic division to quickly check if a particular frequency is a resonant frequency of a circuit, represented by a polynomial equation. If substituting the frequency into the polynomial results in zero (i.e., the remainder is zero in synthetic division), then that frequency is a resonant frequency, which can help in designing filters or amplifiers.

Answered by GinnyAnswer | 2025-07-04