Use the method of equating coefficients to find the values of $a, b$, and $c$: $(x+4)(a x^2+b x+c)=2 x^3+9 x^2+3 x-4$.

A. $a=-2 ; b=1 ; c=-1$
B. $a=2 ; b=-1 ; c=-1$
C. $a=2 ; b=1 ; c=1$
D. $a=2 ; b=1 ; c=-1

By expanding the left side of the equation and equating coefficients, we find the values of a , b , and c are a = 2 , b = 1 , c = − 1 . Therefore, the chosen option is D.
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Answered by Anonymous | 2025-07-04

Expand the left side of the equation: ( x + 4 ) ( a x 2 + b x + c ) = a x 3 + ( b + 4 a ) x 2 + ( c + 4 b ) x + 4 c .
Equate the coefficients: a = 2 , b + 4 a = 9 , c + 4 b = 3 , 4 c = − 4 .
Solve for a , b , and c : a = 2 , c = − 1 , b = 1 .
The solution is: a = 2 , b = 1 , c = − 1 ​

Explanation

Understanding the Problem We are given the equation ( x + 4 ) ( a x 2 + b x + c ) = 2 x 3 + 9 x 2 + 3 x − 4 . Our goal is to find the values of a , b , and c using the method of equating coefficients. This method relies on expanding the left side of the equation and then comparing the coefficients of corresponding terms on both sides.

Expanding the Left Side First, let's expand the left side of the equation:


( x + 4 ) ( a x 2 + b x + c ) = x ( a x 2 + b x + c ) + 4 ( a x 2 + b x + c ) = a x 3 + b x 2 + c x + 4 a x 2 + 4 b x + 4 c = a x 3 + ( b + 4 a ) x 2 + ( c + 4 b ) x + 4 c

Equating Coefficients Now, we equate the coefficients of the corresponding terms on both sides of the equation:

a x 3 + ( b + 4 a ) x 2 + ( c + 4 b ) x + 4 c = 2 x 3 + 9 x 2 + 3 x − 4
This gives us the following system of equations:

a = 2

b + 4 a = 9

c + 4 b = 3

4 c = − 4

Solving the System of Equations Now, let's solve the system of equations for a , b , and c .


From the first equation, we have a = 2 .
From the fourth equation, we have 4 c = − 4 , so c = − 1 .
Substitute a = 2 into the second equation: b + 4 ( 2 ) = 9 , so b + 8 = 9 , which means b = 9 − 8 = 1 .
Substitute b = 1 into the third equation: c + 4 ( 1 ) = 3 , so c + 4 = 3 , which means c = 3 − 4 = − 1 . This confirms our earlier result for c .

Final Values Thus, we have found the values: a = 2 , b = 1 , c = − 1 .

Examples
Equating coefficients is a powerful technique used in various fields, such as electrical engineering, to analyze circuits. For example, when analyzing a circuit with multiple components, you might end up with a polynomial equation describing the voltage or current. By equating coefficients, engineers can determine the values of unknown components or parameters in the circuit, ensuring it functions as designed. This method helps in designing filters, amplifiers, and other essential electronic systems.

Answered by GinnyAnswer | 2025-07-04