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PRACTICE QUESTIONS, CLASS IX : CHAPTER - 2, POLYNOMIALS, , , , , , 1. Factorize the following: 9x” + 6x + 1 — 25y’., , 2. Factorize the following: a° + b> + 2ab + 2be + 2ca, , 3. Show that p(x) = x* — 3x” + 2x — 6 has only one real zero., , 4. Find the value of a if x + 6 is a factor of x° + 3x° + 4x +a., , 5. If polynomials ax’ + 3x” — 3 and 2x* — 5x +a leaves the same remainder when each is divided by, x —4, find the value of a.., , 6. The polynomial f(x)= x‘ — 2x +3x° — ax + b when divided by (x — 1) and (x + 1) leaves the, remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when, f(x) is divided by (x — 2)., , 7. If the polynomials 2x’ +ax” + 3x — 5 and x° + x — 2x +a leave the same remainder when divided, , by (x — 2), find the value of a. Also, find the remainder in each case., , 8. If the polynomials az’ + 42° + 3z — 4 and z’ — 4z + a leave the same remainder when divided by, z~3, find the value of a., , 9. The polynomial p(x) = x" — 2x° + 3x° — ax + 3a — 7 when divided by x + | leaves the remainder, 19. Find the values of a. Also find the remainder when p(x) is divided by x + 2, , , , 1, , 1 2, . both x ~ 2 and x ~ = are factors of px? + 5x + r, show that p = r., , , , , , , , , , 11. Without actual division, prove that 2x* — 5x x + 2 is divisible by x* — 3x + 2, 12. Simplify (2x — 5y)* — (2x + 5y)', 13. Multiply x? + 4y + 2xy + xz — 2yz by (— z +x 2y)., ep, 14. If a, b,c are all non-zero and a +b +c = 0, prove that —+7-+— =3, be ca ab, 15. Ifa+b +c =5 and ab + be + ca = 10, then prove that a’ + b° +c’ -3abe = ~ 25, , 16. Without actual division, prove that 2x" — 6x’ +3x? +3x — 2 is exactly divisible by x” — 3x + 2., 17. Without actual division, prove that x° — 3x” — 13x + 15 is exactly divisible by x” + 2x — 3., , 18. Find the values of a and b so that the polynomial x* — 10x +ax + b is exactly divisible by (x — 1), as well as (x — 2)., , + 5x" -5x-2., , , , 19, Find the integral zeroes of the polynomial 2, , 1, , , , 20. If (x — 3) and (: } are both factors of ax” + 5x + b, then show that a = b., , 3, , 21. Find the values of a and b so that the polynomial x* + ax* — 7x* +8x + b is exactly divisible by, , (x + 2) as well as (x + 3).