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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Question on Polynomials Class 9 SET - 2

Questions on Polynomials Class 9 SET - 2

Q1. Factorize the following by splitting the middle term:

1. 3x² + 19x + 30
2. \(2\sqrt 2 {x^2} + 9x + 5\sqrt 2 \)
3. 3x² - 13x + 10

Q2. Factorize the following by factor theorem:

1. x³ + 9x² + 23x + 15
2. x³ + 6x² + 11x + 6

Q3. Factorize the following by using a suitable identity:

1. 4x² + 12xy + 9y²
2. 2a⁵ - 54a²
3. \(2\sqrt 2 {x^3} + 3\sqrt 3 {y^3}\)
4. x⁵ - x
5. x⁶ - y⁶
6. (a - b)³ + (b - c)³ + (c - a)³
7. x⁸ - y⁸
8. 27x³ - 135x² + 225x - 125

Q4. Evaluate the following by using a suitable identity:

1. 998³
2. (10.2)³
3. 998² - 4
4. 999² - 1
5. (-25)³ + 10³ + 15³
6. 10.2 × 9.8

Q5. If 5 is a zero of x³ + 5x² + 2x + 8, find k.

Q6. If (x - 2) is a zero of x³ - 4x² + kx - 8, find k.

Q7. If (x - 2) and (x + 3) are factors of x³ + ax² + bx - 30, find a and b.

Q8. If x + y + z = 8 and xy + yz + zx = 20, find the value of x³ + y³ + z³ - 3xyz.

Q9. If a + b + c = 9 and a² + b² + c² = 35, find the value of a³ + b³ + c³ - 3abc.

Q10. Find the value of (2.7)³ - (1.6)³ - (1.1)³ using a suitable identity.

Q11. Factorize the following by using a suitable identity:

1. a³ + b³ - 8c³ +6abc
2. \({\left( {\dfrac{a}{b}} \right)^3} + {\left( {\dfrac{b}{c}} \right)^3} + {\left( {\dfrac{c}{a}} \right)^3} - 3\)
3. 8x³ - 27y³ + 125z³ + 90xyz

Q12. Find the value of a³ + 8b³, if a + 2b = 10 and ab = 15.

Q13. Find the value of a³ + 27b³, if a - 3b = - 6 and ab = - 10.

Q14. find the value of m and n, if y² -1 is a factor of y⁴ + my³ + 2y² - 3y + n.

Q15. (x + 2) is a factor of mx³ + nx² + x - 6. it leaves the remainder 4 when divided by (x - 2). find m and n. 

Q16. The polynomials kx³ + 3x² - 3 and 2x³ - 5x + k leave the same remainder when divided by (x - 4). Find k.

Q17. Factorize: \({x^2} + \dfrac{1}{{{x^2}}} - 2\).

QQ18. If ab + bc + ca = 10 and a² + b² + c² = 44. find a³ + b³ + c³ - 3abc.

Q19. Show that x - 2 is a factor p(x) = x³ - 12x² + 44x - 48.

Q20. Factorize: x³ - 2x² - 5x + 6.

Q21. If x + p is a factor of p(x) = x⁵ - p²x³ + 44x - 40. Find p.

Q22. Factorize:

1. \({a^2} + \dfrac{1}{{{a^2}}} + 2\)
2. 3(x + 2)² + 17(x + 2) + 10
3. 4x² - 9y² + 20x + 25

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Questions on Polynomials class 9 SET -1

Extra Questions on Polynomials Class 9

Q1. Find the remainder when \({y^3} + {y^2} - 2y + 5\) is divided by y - 5.

Q2. Determine the remainder when p(x) = \({x^3} + 3{x^2} - 6x + 15\) is divided by x - 2.

Q3.  When \(f(x) = {x^4} - 2{x^3} + 3b{x^2} - ax\) is divided by x+1 and x - 1, we get  remainder as 19 and 5 respectively. Find the remainder if f(x) is divided by x - 3.

Q4.  What must be subtracted from \(4{x^4} - 2{x^3} - 6{x^2} + x - 5\) so that the result is exactly divisible by \(2{x^2} + 3x - 2\) ?

Q5.  If (x + 1) and (x - 1) both are factors of \(a{x^3} + {x^2} - 2x + b\), find a and b.

Q6.  Factorize each of the following expressions:

1. \(48{x^3} - 36{x^2}\)
2. \(5{x^2} - 15xy\)
3. \(15{x^3}{y^2}z - 25x{y^2}{z^2}\)

Q7. Factorize:

1. \(2{x^2}(x + y) - 3(x + y)\)
2. \(5xy(5x + y) - 5y(5x + y)\)
3. \(x({x^2} + {y^2} - {z^2}) + y({x^2} + {y^2} - {z^2}) + z({x^2} + {y^2} - {z^2})\)
4. \(ab({a^2} + {b^2} - {c^2}) + bc({a^2} + {b^2} - {c^2}) + ca({a^2} + {b^2} - {c^2})\)

Q8. Factorize each of the following expressions:

1. \(25{x^2}{y^2} - 20x{y^2}z + 4{y^2}{z^2}\)
2. \(4{x^2} - 4\sqrt 7 x + 7\)
3. \(\dfrac{{{a^2}}}{{{b^2}}} + 2 + \dfrac{{{b^2}}}{{{a^2}}}\)
4. \(4{a^2} + 12ab + 9{b^2} - 8a - 12b\)

Q9. Factorize each of the following:

1. \(25{x^2} - 36{y^2}\)
2. \(2ab - {a^2} - {b^2} + 1\)
3. \(36{a^2} - 12a + 1 - 25{b^2}\)
4. \({a^4} - 81{b^4}\)
5. \({a^{12}}{b^4} - {a^4}{b^{12}}\)
6. \(4{x^2} - 9{y^2} - 2x - 3y\)

Q10. Factorize by completing the square.

1. \({a^4} + {a^2} + 1\)
2. \({y^4} + 5{y^2} + 9\)
3. \({x^4} + 4\)
4. \({x^4} + 4{x^2} + 3\)

Q11. Factorize by completing the square.

1. \({a^3} - 27\)
2. \(1 - 27{x^3}\)
3. \(8{x^3} - {(2x - 3y)^3}\)
4. \({a^8} - {a^2}{b^6}\)
5. \({a^3} - 5\sqrt 5 {b^3}\)

Q12. Factorize the following:

1. \(16{p^3}{q^2} + 54{r^3}\)
2. \(\dfrac{{{a^3}}}{8} + 8{b^3}\)
3. \(2\sqrt 2 {a^3} + 3\sqrt 3 {b^3}\)
4. \(8{a^4}b + \dfrac{1}{{125}}a{b^4}\)
5. \({a^7} - 64a\)

Q13.Q13. Factorize:

1. \({x^3} + 9{x^2} + 27x + 27\)
2. \({x^3} - 9{x^2}y + 27x{y^2} - 27{y^3}\)

Q14. Using identities, find the value of

1. 101²
2. 98²
3. (0.98)²
4. 101 × 99
5. 190 × 190 - 10 × 10

Q15. Expand using suitable identity

1. (x + 5y + 6z)²
2. (2a - 3b + 4c)²
3. ( - a + 6b + 5c)²
4. (- p + 4q - 3r)²

Q16. Expand using suitable identity

1. (2x + 5y)³
2. (5p - 3q)³
3. (- a + 2b)³

Q17. Evaluate using identities

1. 102³
2. 99³

Q18. Simplify : 

1. (2a + b)³ + (2a - b)³
2. (4x + 5y)³ - (4x - 5y)³

Q19. Factorize:

1. 30x³y + 24x²y² - 6xy
2. 5x(a - b) + 6y(a - b)

Q20. Factorize:

1. 9x² - y²
2. (3 - x)² - 36x²
3. (2x - 3y)² - (3x + 4y)²
4. 16x⁴ - y⁴

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