Polynomials important questions

POLYNOMIALS

1. If the squared difference of the zeros of the quadratic polynomial f(x) = x² + px + 45 is equal to 144, find the value of p.
2. If α and β are the zeros of the quadratic polynomial f(x) = x² – px + q, prove that
   
   (α²/β²) + (β²/α²) = (p⁴/q²) – (4p²/q) + 2.
3. If α and β are the zeros of the quadratic polynomial f(x) = x² – p(x + 1) – c, show that
   
   (α + 1)(β + 1) = 1 – c.
4. If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α – β = 8, find a quadratic polynomial having α and β as its zeros.
5. If α and β are the zeros of the quadratic polynomial f(x) = x² – 1, find a quadratic polynomial whose zeros are (2α/β) and (2β/α).
6. If α and β are the zeros of the quadratic polynomial f(x) = x² – 3x – 2, find a quadratic polynomial whose zeros are 1/(2α + β) and 1/(2β + α).
7. If α and β are the zeros of the polynomial f(x) = x² + px + q, form a polynomial whose zeros are (α + β)² and (α – β)².
8. If α and β are the zeros of the quadratic polynomial f(x) = x² – 2x + 3, find a polynomial whose roots are:
   
   (i) α + 2, β + 2
   
   (ii) (α – 1)/(α + 1), (β – 1)/(β + 1)
9. If α and β are the zeros of the quadratic polynomial f(x) = ax² + bx + c, then evaluate:
   
   (i) α – β
   
   (ii) 1/α – 1/β
   
   (iii) 1/α + 1/β – 2αβ
   
   (iv) α²β + αβ²
   
   (v) α⁴ + β⁴
   
   (vi) 1/(aα + b) + 1/(aβ + b)
   
   (vii) β/(aα + b) + α/(aβ + b)
   
   (viii) a(α²/β + β²/α) + b(α/β + β/α)
10. For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization:
    
    (i) 8/3, 4/3
    
    (ii) 21/8, 5/16
    
    (iii) -2√3, -9
    
    (iv) -3/(2√5), 1/2
11. If α and β are the zeros of the quadratic polynomial f(x) = x² – 5x + 4, find the value of
    
    1/α + 1/β – 2αβ.
12. If α and β are the zeros of the quadratic polynomial p(y) = 5y² – 7y + 1, find the value of
    
    1/α + 1/β.
13. If α and β are the zeros of the quadratic polynomial f(x) = x² – x – 4, find the value of
    
    1/α + 1/β – αβ.
14. If α and β are the zeros of the quadratic polynomial f(x) = x² + x – 2, find the value of
    
    1/α – 1/β.
15. If one zero of the quadratic polynomial f(x) = 4x² – 8kx – 9 is negative of the other, find the value of k.
16. If the sum of the zeroes of the quadratic polynomial f(t) = kt² + 2t + 3k is equal to their product, find the value of k.

LEVEL–2

17. If α and β are the zeros of the quadratic polynomial p(x) = 4x² – 5x – 1, find the value of α²β + αβ².
18. If α and β are the zeros of the quadratic polynomial f(t) = t² – 4t + 3, find the value of α⁴β³ + α³β⁴.
19. If α and β are the zeros of the quadratic polynomial f(x) = 6x² + x – 2, find the value of
    
    α/β + β/α.
20. If α and β are the zeros of the quadratic polynomial p(s) = 3s² – 6s + 4, find the value of
    
    α/β + β/α + 2(1/α + 1/β) + 3αβ.