A cubic polynomial f( x )= ax 3+ bx 2+ cx + d has a graph which is tangent to the x-axis at 2 , has another x-intercept at -1 , and has y-intercept at -2 as shown. The value of, a + b + c + d equals

Question

A cubic polynomial f( x )= ax 3+ bx 2+ cx + d has a graph which is tangent to the x-axis at 2 , has another x-intercept at -1 , and has y-intercept at -2 as shown. The value of, a + b + c + d equals (A) -2 (B) -1 (C) 0 (D) 1

Tardigrade Admin · Accepted Answer

The polynomial must be of the form f(x)=a(x-2)2(x +1) because it has a double zero at 2 and a zero at -1 . To solve for ' a ', note that f (0)= a (0-2)2(0+1)=-2. It follows that a =-1 / 2 Hence f(x)=-(1/2)[x3-3 x2+4] ∴ a =-(1/2), b =(3/2), c =0 ; d =-2 ⇒ c + b + c + d =-1

Q. A cubic polynomial f(x)=ax3+bx2+cx+d has a graph which is tangent to the x-axis at 2 , has another x-intercept at -1 , and has y-intercept at -2 as shown. The value of, a+b+c+d equals

-2

-1

0

1

The polynomial must be of the form f(x)=a(x−2)2(x +1) because it has a double zero at 2 and a zero at -1 . To solve for ' a ', note that f(0)=a(0−2)2(0+1)=−2. It follows that a=−1/2 Hence f(x)=−21​[x3−3x2+4] ∴a=−21​,b=23​,c=0;d=−2 ⇒c+b+c+d=−1

Q. A cubic polynomial $f (x) = a x^{3} + b x^{2} + c x + d$ has a graph which is tangent to the $x$ -axis at 2 , has another $x$ -intercept at -1 , and has y-intercept at -2 as shown. The value of, $a + b + c + d$ equals