Question

If the quadratic polynomial \((x^2-x-2)\) is a factor of the polynomial \((x^4-x^3+ax^2+bx+4)\), then \((a,b)=\)

• A. \((4,4)\)
• B. \((-2,-3)\)
• C. \((-3,-2)\)
• D. \((-4,2)\)

Hint:

If a polynomial factor is given, first find its roots and then substitute those roots into the larger polynomial to form linear equations.

Answer 1

The Correct Option is B

Step 1: Factorize the quadratic factor. \[ x^2-x-2=(x-2)(x+1) \] Hence \(x=2\) and \(x=-1\) are roots of \[ f(x)=x^4-x^3+ax^2+bx+4 \]

Step 2: Use \(f(2)=0\). \[ 16-8+4a+2b+4=0 \] \[ 12+4a+2b=0 \] \[ 2a+b=-6 \] \[ \cdots (1) \]

Step 3: Use \(f(-1)=0\). \[ 1+1+a-b+4=0 \] \[ a-b=-6 \] \[ \cdots (2) \]

Step 4: Solve equations (1) and (2). From (2), \[ a=b-6 \] Substituting into (1), \[ 2(b-6)+b=-6 \] \[ 3b=6 \] \[ b=2 \] \[ a=-4 \] Therefore \[ \boxed{(a,b)=(-4,2)} \]