Polynomial - Practice Questions

1. The graph of y = p(x) are drawn below. Find the number of zeros in each case:
2. Check whether the following are polynomials or not. If they are polynomial classify them as linear, quadratic or cubic polynomial. If they are not , give reasons for this. a) p(x) = √x+3 b) q(x) = x³+x-1 c) p(x) = (2/3)x³+1 d) t(x) = x²+5x+1 
3. If α, β are zeroes of the polynomial p(x) = 2x²- 4x+5 , find the value of a) α ²+β ² b) (α- β)² 
4. If the sum of zeroes of the polynomial p(x) =2x³-3ax²+4x-5 is 6 , then find the value of a. 
5. If the product of two zeroes of the polynomial p(x) = 2x³+6x²-4x+9 is 3, then find its third zero. 
6. If one zero of the polynomial p(x) = 5x²+13x-a is reciprocal of the other, find the value of a. 
7. If α, β are zeroes of the polynomial p(x) x²-(k+6)x+2(2k-1) , find k if 2(α+ β) =α β 
8. Find the zeroes of each of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients of the polynomials: a) p(x) = 2x² – 9 – 3x b) q(s) = 2s²+5x+3 c) p(t)=t²-5 d) p(u)=4u²+8u 
9. Show that –1 , 3 and 6 are the zeroes of the polynomial p(x) =x³-8x²+9x+18. Also verify the relationship between the zeroes and the coefficients of p(x). 
10. Find the cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time and product of its zeroes being given below: a) 4 , 1 and –6 b) 1 , -1 and 2 
11. On dividing 3x³-2x²+5x-5 by a polynomial p(x) , the quotient and remainder are x²-x+2 and –7 respectively. Find p(x). 
12. By division algorithm, check whether or not x2+3x+1 is a factor of 3x⁴+5x³-7x²+2x+2 
13. Find the value of b for which the polynomial 2x+3 is a factor of 2x³+9x²-x-b. 
14. What must be suptracted from 4x⁴+2x³-8x²+3x-7 so that it may be exactly divisible by 2x²+x-2? 
15. If α , β , γ are the zeros the polynomial 6x³+3x²-5x+1 , then find the value of α^-1+β^-1+γ ^-1 
16. If the polynomial 6x⁴+8x³+17x²+21x+7 is divided by another polynomial 3x²+4x+1 , the remainder comes out to be (ax+b) , find a and b.
17. Find all the zeroes of the polynomial 2x⁴+7x³-19x²-14x+30 , if two of its zeroes are √2 and -√2 
18. If one zero of the polynomial (a²+9)x²+13x+6a is reciprocal of the other. Find the value of a. 
19. Find the zeroes of a polynomial f(x) = x³-5x²-16x+80 , if its two zeroes are equal in magnitude but opposite in sign. 
20. If α , β are zeroes of a quadratic polynomial such that α + β = 24 and α - β = 8 , find the quadratic polynomial having α , β as its zeroes.