Polynomials

Polynomials

1. Find the zeroes of the polynomial f(x) = x² + 7x +12 and verify the relation between its zeroes and coefficients.
   
2. Find the quadratic polynomial whose zeroes are 2/3 and -1/4.
   
3. Find the quadratic polynomial whose zeroes are 2 and -6.
   
4. Find the quadratic polynomial, the sum and product of whose zeroes are √2 and -12 resp.
   
5. Find the zeroes of the quadratic polynomial (x² – 5) and verify the relation between its zeroes and its coefficients.
   
6. Verify that 2, -3 and 1/2 are the zeroes of the cubic polynomial f(x) = 2x³ + x² - 13x + 6 and then verify the relation between its zeroes and coefficients.
   
7. Find a cubic polynomial whose zeroes are 3,5 and -2.
   
8. 4x³ - 8x² + 8x + 1 when divided by g(x) gives (2x-1) as quotient and (x+3) as remainder. Find g(x).
   
9. Find all the zeroes of the polynomial f(x)=2x⁴ - 3x³ - 5x² + 9x – 3, it being given that two of its zeroes are √3 and -√3.
   
10. Divide 12-17x-5x² by 3-5x and verify the division algorithm.
    
11. It being given that 1 is one of the zeroes of the polynomial 7x - x³ -6. Find its other zeroes.
    
12. If the polynomial x⁴ - 6x³ + 16x² - 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a. Find the values of k and a.
    
13. If the zeroes of the polynomial x³ - 3x² + x +1 are a-b, a and a=b, find the values of a and b.
    
14. Give examples of polynomials p(x), g(x), q(x) and r(x), where p(x) is divided by g(x) and q(x) and r(x) are quotient and remainder resp. which satisfy the division algorithm and
    

    a. deg p(x) = deg r(x)      b. deg q(x) = deg r(x)            c. deg r(x) =0                 d. r(x) = 0

15. If x + a is a factor of 2x³ - 3x² + x +10, find a.
    
16. If one zero of the polynomial (a² + 9)x² + 13x +6a is reciprocal of the other, find the value of a.
    
17. If the product of the zeroes of the polynomial ax² - 6x - 6 is 4, find the value of a.