Polynomial Equation

Question

(a) Three factors of x⁴ +ax³ +bx² +x +c are x, x+1 and x-1. Find a, b and c.

(b) Given m(x) = x³ +px² +qx +r. When m(x) is divided by x, x-2 and x+3, the remainder is 4. Find p, q and r

Answer 1

a)  if x is a factor of the stated 4th degree polynomial, then c=0.

if x+1 is a factor of f(x) = x⁴+ax³+bx²+x then  f(-1) = 0; this means 0 = 1 - a + b 

if x-1 is a factor of f(x) then f(1) = 0 so 0 = 1 +a + b + 1.  These two equations are based on the Remainder Theorem.

You can now solve for a and b.  Remember to check your work.

 

b) Similar work using Remainder Theorem will lead you to a solution of this second problem.