Questions on polynomials Class 9 SET-3

Q1. Find the zeros of the polynomial

1. x² + 7x +  10
2. x² - 25

Q2. Factorize: 

1. 999² - 1
2. (10.2)³
3. 1002 × 998

Q3. Factorize:

1. x³ - 3x² - 9x - 5
2. x³ + 7x² - 21x - 27

Q4. Factorize:

1. \(3{x^2} + 27{y^2} + {z^2} - 18xy + 6\sqrt 3 yz - 2\sqrt 3 zx\)
2. 27x³ + 125y³
3. \(\dfrac{1}{{64}}{a^2} + {b^3} + 125{c^3} - \dfrac{{15}}{4}abc\)
4. \({x^3} - \dfrac{1}{{{x^3}}}\)
5. 8x³ - (2x - y)³
6. a⁶ - b⁶

Q5. Using factor theorem, show that (a - b) is the factor of a(b² - c²) + b(c² - a²) + c(a² - b²).

Q6. Factorize:

1. \(4\sqrt 3 {x^2} + 5x - 2\sqrt 3 \)
2. \(21{x^2} - 2x + \dfrac{1}{{21}}\)
3. 9(2a - b)² - 4(2a - b) - 13

Q7. Simplify and factorize (a + b + c)² - (a - b - c)² + 4b² - 4c² 

Q8. Factorize: (a² - b²)³ + (b² - c²)³ + (c² - a²)³

Q9. For what value of a is 2x³ + ax² + 11x + a + 3 exactly divisible by (2x - 1).

Q10. If x  - 2 is a factor of a polynomial f(x) = x⁵ - 3x⁴ - ax³ + 2ax + 4, then find the value of a.

Q11. Find the value of a and b so that x² - 4 is a factor of ax⁴ + 2x³ - 3x² + bx - 4

Q12. If x = 2 and x = 0 are zeroes of the polynomial 2x³ - 3x² + px + q, then find the value of p and q.

Q13. Find the value of a and b, so that x³ - ax² - 13x + b is exactly divisible by (x - 1) as well as (x + 3).

Q14. The polynomial x³ - mx² + 4x + 6 when divided by (x + 2) leaves remainder 14, find m.

Q15. If the polynomial ax³ + 3x² - 13 and 2x³  - 15x + a, when divided by (x - 2)  leave the same remainder. Find the value of a.

Q16. If both (x -  2) and \(\left( {x - \dfrac{1}{2}} \right)\)  are the factors of px² + 5x + r, show that p = r.

Q17. If f(x) = x⁴ - 2x³ + 3x² - ax + b is divided by x - 1 and x  + 1 the remainders are 5 and 19 respectively, then find a and b.

Q18. If A and B be the remainders when the polynomials x³ + 2x² - 5ax - 7 and x³ + ax² - 12x + 6 are divided by (x + 1) and (x - 2) respectively and 2A + B = 6, find the value of a.

Q19. Show that x + 1 and 2x - 3 are factors of 2x³ - 9x² + x + 12.

Q20. If sum of remainders obtained by dividing ax³ - 3ax² + 7x + 5 by (x + 1) is -36. find a.

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