Question

Use the method of equating coefficients to find the values5

Answer 1

Given: The equation below

`(x+4)(ax^2+bx+c)=2x^3+9x^2+3x-4`

To Determine: The value of a, b, and c, using the method of equating coefficients

Solution

Let us expand the left hand side of the equation

`\begin{gathered} (x+4)(ax^2+bx+c) \\ =ax^3+bx^2+cx+4ax^2+4bx+4c \end{gathered}`

Let us collect like terms

`\begin{gathered} =ax^3+bx^2+4ax^2+cx+4bx+4c \\ =ax^3+(b+4a)x^2+(c+4b)x+4c \end{gathered}`

Let us comapre the coefficients of the left hand side and the right hand side

`ax^3+(b+4a)x^2+(c+4b)x+4c=2x^3+9x^2+3x-4`
`\begin{gathered} coefficient(x^3) \\ a=2 \end{gathered}`
`\begin{gathered} coefficient(x^2) \\ b+4a=9 \\ coefficient(x) \\ c+4b=3 \\ constant \\ 4c=-4 \\ c=-(4)/(4) \\ c=-1 \end{gathered}`

Note

`\begin{gathered} b+4a=9 \\ a=2 \\ b+4(2)=9 \\ b+8=9 \\ b=9-8 \\ b=1 \end{gathered}`

Hence

a = 2

b = 1

c = -1

The correct option is OPTION C