Question

If \(ax+b\) is added to \(4x^4+2x^3-2x^2+x-1\), then it is divisible by \(x^2+2x-3\). Then \(2a+b=\)

• A. 61
• B. 57
• C. 51
• D. 46

Hint:

If a polynomial is divisible by \((x-r)\), then substituting \(x=r\) makes the polynomial equal to zero.

Answer 1

The Correct Option is B

Concept: If a polynomial is divisible by \((x+3)(x-1)\), then it must become zero at \(x=1\) and \(x=-3\).

Step 1: Forming the polynomial.
\[ P(x)=4x^4+2x^3-2x^2+x-1+ax+b \] \[ =4x^4+2x^3-2x^2+(a+1)x+(b-1) \]

Step 2: Using \(P(1)=0\).
\[ 4+2-2+a+1+b-1=0 \] \[ a+b+4=0 \] \[ a+b=-4 \]

Step 3: Using \(P(-3)=0\).
\[ 4(81)+2(-27)-2(9)-3(a+1)+(b-1)=0 \] \[ 324-54-18-3a-3+b-1=0 \] \[ 248-3a+b=0 \] \[ b=3a-248 \] Substituting in \(a+b=-4\), \[ a+3a-248=-4 \] \[ 4a=244 \] \[ a=61 \] \[ b=-65 \]

Step 4: Finding \(2a+b\).
\[ 2a+b=2(61)-65 \] \[ =122-65 \] \[ =57 \] {57}