General

Very Hard Polynomial Questions

It's not the very very basic things you learn with polynomials. A little harder.

1. Find all zeros of f(x)=128x^3-48x^2+1 given that one linear factor occurs twice.

2. Find all zeros of f(x)=x^4-54x^2+216x-243

3. Find the relationship between p and q such that the zeros of the polynomial x^2+px+q are in the ratio 3:2

4. Find the value of m =/= 0 if one of the zeros of x^2+(m+1) is three times the other.

5. Find the values of p if the polynomial x^2+px+2
a. has 2 as one of its zeros;
b. has zeros whose sum is twice their product;
c. has zeros whose squares have a sum of 21;
d. has zeros whose difference is 3/2.

6. Use the remainder theorem to show that x+a+b+c is a factor of the polynomial f(x) = (x+a)(x+b)(x+c)+(b+c)(c+a)(a+b). Hence solve the equation (x+2)(x-3)(x-1)+4=0

7. Find the zeros of the polynomial x^3-9x^2+11x+21 if the zeros form an arithmetic progression.

8. If x-3 is a factor of the polynomial f(x)=x^5-x^4+a, show that x-3 is a factor of the polynomial g(x)=-2(x-6)^4-a

9. When a polynomial P(x) is divided by x-2, x and x+2, the remainders are 9, 3, and 13 respectively. Find the remainder when P(x) is divided by x(x^2-4)

10. When a polynomial f(x) is divided by x^2-a^2 (a=/=0), the remainder is px+q.
a. Prove that p=1/(2a)[f(a)-f(-a)] and q = 1/2[f(a)+f(-a)]
b. Find the remainder when x^n-a^n is divded by x^2-a^2 when
i. n is even
ii. n is odd