Answer: A=2 and B=13 Explanation: The Factor Theorem states that if a is the root of any polynomial p(x) that is if p(a)=0, then (x−a) is the factor of the polynomial p(x).
Let p(x)=x 3 +ax 2 −bx+10 and g(x)=x 2 −3x+2 Factorise g(x)=x 2 −3x+2: x 2 −3x+2=x 2 −2x−x+2=x(x−2)−1(x−2)=(x−2)(x−1) Therefore, g(x)=(x−2)(x−1) It is given that p(x) is divisible by g(x), therefore, by factor theorem p(2)=0 and p(1)=0. Let us first find p(2) and p(1) as follows: p(1)=1 3 +(a×1 2 )−(b×1)+10=1+(a×1)−b+10=a−b+11 p(2)=2 3 +(a×2 2 )−(b×2)+10=8+(a×4)−2b+10=4a−2b+18 Now equate p(2)=0 and p(1)=0 as shown below: a−b+11=0 ⇒a−b=−11.......(1) 4a−2b+18=0 ⇒2(2a−b+9)=0 ⇒2a−b+9=0 ⇒2a−b=−9.......(2) Now subtract equation 1 from equation 2:
(2a−a)+(−b+b)=(−9+11) ⇒a=2 Substitute a=2 in equation 1: 2−b=−11 ⇒−b=−11−2 ⇒−b=−13 ⇒b=13 Hence, a=2 and b=13.