Question

Use the method of equating coefficients to find the values of a, b, and c: (x + 4) (ar²+bx+c) = 2x³ + 9x² + 3x - 4.A. a = -2; b= 1; c= -1OB. a=2; b= 1; c= 1OC. a=2; b= -1; c= -1OD. a=2; b= 1; c= -1

Answer 1

To find the coefficients we first need to make the multipliation on the left expression:

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

Then we have:

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

Two polynomials are equal if and only if their coefficients are equal, this leads to the following equations:

`\begin{gathered} a=2 \\ 4a+b=9 \\ 4b+c=3 \\ 4c=-4 \end{gathered}`

From the first one it is clear that the value of a is 2, from the last one we have:

`\begin{gathered} 4c=-4 \\ c=-(4)/(4) \\ c=-1 \end{gathered}`

Plugging the value of a in the second one we have:

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

Therefore, we conclude that a=2, b=1 and c=-1 and the correct choice is D.