Use synthetic division to find the remainder if polynomial [tex]P(x)=x^3-28 x-48[/tex] is divided by [tex]x+4[/tex].
A) 0
B) 3
C) 2
D) 1
Answer (1)
Set up synthetic division with the root of the divisor (-4) and the coefficients of the polynomial (1, 0, -28, -48).
Perform synthetic division.
The remainder is the last number in the bottom row, which is 0.
Therefore, the remainder is 0 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = x 3 − 28 x − 48 and asked to find the remainder when it is divided by x + 4 using synthetic division.
Setting up Synthetic Division The divisor is x + 4 , so we want to find P ( − 4 ) . We can use synthetic division or direct substitution to find the remainder. Let's use synthetic division.
Performing Synthetic Division We set up the synthetic division with -4 (the root of x + 4 ) and the coefficients of P ( x ) , which are 1, 0, -28, and -48. Note that we include a 0 for the x 2 term since it is missing in the polynomial.
Finding the Remainder
1
0
-28
-48
-4
-4
16
48
1
-4
-12
0
The last number in the bottom row is the remainder, which is 0.
Direct Substitution Verification Alternatively, we can directly substitute x = − 4 into the polynomial:
P ( − 4 ) = ( − 4 ) 3 − 28 ( − 4 ) − 48 = − 64 + 112 − 48 = 0
Thus, the remainder is 0.
Final Answer The remainder when P ( x ) = x 3 − 28 x − 48 is divided by x + 4 is 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 fields like engineering and physics, where polynomial equations frequently arise when modeling physical systems. For example, when analyzing the stability of a control system, engineers often need to find the roots of a characteristic polynomial. Synthetic division can quickly help determine if a particular value is a root, simplifying the analysis and design process. It allows for efficient determination of whether a specific input will lead to system instability or resonance, ensuring safer and more reliable designs.
