Chapters
Chapter 2: Exponents of Real Numbers
Chapter 3: Rationalisation
Chapter 4: Algebraic Identities
Chapter 5: Factorisation of Algebraic Expressions
Chapter 6: Factorisation of Polynomials
Chapter 7: Linear Equations in Two Variables
Chapter 8: Co-ordinate Geometry
Chapter 9: Introduction to Euclid’s Geometry
Chapter 10: Lines and Angles
Chapter 11: Triangle and its Angles
Chapter 12: Congruent Triangles
Chapter 13: Quadrilaterals
Chapter 14: Areas of Parallelograms and Triangles
Chapter 15: Circles
Chapter 16: Constructions
Chapter 17: Heron’s Formula
Chapter 18: Surface Areas and Volume of a Cuboid and Cube
Chapter 19: Surface Areas and Volume of a Circular Cylinder
Chapter 20: Surface Areas and Volume of A Right Circular Cone
Chapter 21: Surface Areas and Volume of a Sphere
Chapter 22: Tabular Representation of Statistical Data
Chapter 23: Graphical Representation of Statistical Data
Chapter 24: Measures of Central Tendency
Chapter 25: Probability
Chapter 6: Factorisation of Polynomials
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.1 [Pages 2 - 3]
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
3x2 - 4x +15
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
`y^2 +2sqrt3`
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
`3sqrtx+sqrt2x`
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
`x - 4/x`
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
`x^12+y^3+t^50`
Write the coefficient of x2 in the following:
`17 -2x + 7x^2`
Write the coefficient of x2 in the following:
`9-12x +X^3`
Write the coefficient of x2 in the following:
`pi/6x^2- 3x+4`
Write the coefficient of x2 in the following:
`sqrt3x-7`
Write the degrees of each of the following polynomials
`7x3 + 4x2 – 3x + 12`
Write the degrees of the following polynomials:
`12-x+2x^3`
Write the degrees of the following polynomials:
`5y-sqrt2`
Write the degrees of the following polynomials:
7
Write the degrees of the following polynomials
0
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
`x+x^2 +4`
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
`3x-2`
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
`2x+x^2`
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
`3y`
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:
`t^2+1`
Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials
`7t^4+4t^3+3t-2`
Classify the following polynomials as polynomials in one-variable, two variables etc:
`x^2-xy+7y^2`
Classify the following polynomials as polynomials in one-variable, two variables etc:
`x^2-2tx+7t^2-x+t`
Classify the following polynomials as polynomials in one-variable, two variables etc:
`t^3_3t^2+4t-5`
Classify the following polynomials as polynomials in one-variable, two variables etc:
`xy+yx+zx`
Identify polynomials in the following:
`f(x)=4x^3-x^2-3x+7`
Identify polynomials in the following:
`g(x)=2x^3-3x^2+sqrtx-1`
Identify polynomials in the following:
`p(x)=2/3x^3-7/4x+9`
Identify polynomials in the following:
`q(x)=2x^2-3x+4/x+2`
Identify polynomials in the following:
`h(x)=x^4-x^(3/2)+x-1`
Identify polynomials in the following:
`f(x)=2+3/x+4x`
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
`f(x)=0`
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
`g(x)=2x^3-7x+4`
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
`h(x)=-3x+1/2`
Identify constant, linear, quadratic and cubic polynomials from the following polynomials
`p(x)=2x^2-x+4`
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
`q(x)=4x+3`
Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
`r(x)=3x^2+4x^2+5x-7`
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.2 [Page 8]
If `f(x)=2x^2-13x^2+17x+12` find `f(2)`
If `f(x)=2x^2-13x^2+17x+12` find `f-(3)`
If `f(x)=2x^2-13x^2+17x+12` find `f(0)`
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`f ( x ) = 3x +1, x = - 1/3`
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`f(x)=x^2- 1,x=1,-1`
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`g(x)=3x^2-2,` `x=2/sqrt3 2/sqrt3`
Verify whether the indicated numbers is zeros of the polynomials corresponding to them in the following case:
\[p(x) = x^3 - 6 x^2 + 11x - 6, x = 1, 2, 3\]
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`f ( x ) = 5x - pi , x = 4/5`
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`f ( x) = x^2and x = 0`
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`f(x) = lx + m , x = - m/1`
Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:
`f (x) = 2x +1, x = 1/2`
If `x = 2` is a root of the polynomial `f(x) = 2x2 – 3x + 7a` find the value of a.
If `x = −1/2` is a zero of the polynomial `p(x)=8x^3-ax^2 -+2` find the value of a.
If x = 0 and x = −1 are the roots of the polynomial f(x) =2x3 − 3x2 + ax + b, find the value of a and b.
Find the integral roots of the polynomial f(x) = x3 + 6x2 + 11x + 6.
Find rational roots of the polynomial f(x) = 2x3 + x2 − 7x − 6.
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.3 [Pages 14 - 15]
In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the result by actual division: (1−8)
f(x) = x3 + 4x2 − 3x + 10, g(x) = x + 4
f(x) = 4x4 − 3x3 − 2x2 + x − 7, g(x) = x − 1
f(x) = 2x4 − 6x3 + 2x2 − x + 2, g(x) = x + 2
f(x) = 4x3 − 12x2 + 14x − 3, g(x) 2x − 1
f(x) = x3 − 6x2 + 2x − 4, g(x) = 1 − 2x
f(x) = x4 − 3x2 + 4, g(x) = x − 2
f(x) = 9x3 − 3x2 + x − 5, g(x) = \[x - \frac{2}{3}\]
\[f(x) = 3 x^4 + 2 x^3 - \frac{x^2}{3} - \frac{x}{9} + \frac{2}{27}, g(x) = x + \frac{2}{3}\]
If the polynomials 2x3 + ax2 + 3x − 5 and x3 + x2 − 4x +a leave the same remainder when divided by x −2, find the value of a.
If the polynomials ax3 + 3x2 − 13 and 2x3 − 5x + a, when divided by (x − 2) leave the same remainder, find the value of a.
Find the remainder when x3 + 3x2 + 3x + 1 is divided by x + 1.
Find the remainder when x3 + 3x2 + 3x + 1 is divided by \[x - \frac{1}{2}\].
Find the remainder when x3 + 3x2 + 3x + 1 is divided by x.
Find the remainder when x3 + 3x2 + 3x + 1 is divided by \[x + \pi\] .
Find the remainder when x3 + 3x2 + 3x + 1 is divided by 5 + 2x .
The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the value of the following case, if R1 = R2.
The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the values of the following case, if R1 + R2 = 0.
The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the values of the following cases, if 2R1 − R2 = 0.
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.4 [Pages 24 - 25]
In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not: (1−7)
f(x) = x3 − 6x2 + 11x − 6; g(x) = x − 3
f(x) = 3x4 + 17x3 + 9x2 − 7x − 10; g(x) = x + 5
f(x) = x5 + 3x4 − x3 − 3x2 + 5x + 15, g(x) = x + 3
f(x) = x3 −6x2 − 19x + 84, g(x) = x − 7
f(x) = 3x3 + x2 − 20x +12, g(x) = 3x − 2
f(x) = 2x3 − 9x2 + x + 12, g(x) = 3 − 2x
f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2
Show that (x − 2), (x + 3) and (x − 4) are factors of x3 − 3x2 − 10x + 24.
Show that (x + 4) , (x − 3) and (x − 7) are factors of x3 − 6x2 − 19x + 84
For what value of a is (x − 5) a factor of x3 − 3x2 + ax − 10?
Find the value of a such that (x − 4) is a factors of 5x3 − 7x2 − ax − 28.
Find the value of a, if x + 2 is a factor of 4x4 + 2x3 − 3x2 + 8x + 5a.
Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.
Find the values of a and b, if x2 − 4 is a factor of ax4 + 2x3 − 3x2 + bx − 4
Find α and β, if x + 1 and x + 2 are factors of x3 + 3x2 − 2αx + β.
If x − 2 is a factor of the following two polynomials, find the values of a in each case x3 − 2ax2 + ax − 1.
If x − 2 is a factor of the following two polynomials, find the values of a in each case x5 − 3x4 − ax3 + 3ax2 + 2ax + 4.
In the following two polynomials, find the value of a, if x − a is factor x6 − ax5 + x4 − ax3 + 3x − a + 2.
In the following two polynomials, find the value of a, if x − a is factor (x5 − a2x3 + 2x + a + 1).
In the following two polynomials, find the value of a, if x + a is a factor x3 + ax2 − 2x +a + 4.
In the following two polynomials, find the value of a, if x + a is a factor x4 − a2x2 + 3x −a.
Find the values of p and q so that x4 + px3 + 2x3 − 3x + q is divisible by (x2 − 1).
Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.
If x3 + ax2 − bx+ 10 is divisible by x2 − 3x + 2, find the values of a and b
If both x + 1 and x − 1 are factors of ax3 + x2 − 2x + b, find the values of a and b.
What must be added to x3 − 3x2 − 12x + 19 so that the result is exactly divisibly by x2 + x - 6 ?
What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?
What must be added to 3x3 + x2 − 22x + 9 so that the result is exactly divisible by 3x2 + 7x − 6?
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.5 [Pages 32 - 33]
Using factor theorem, factorize each of the following polynomials:
x3 + 6x2 + 11x + 6
x3 + 2x2 − x − 2
x3 − 6x2 + 3x + 10
x4 − 7x3 + 9x2 + 7x − 10
3x3 − x2 − 3x + 1
x3 − 23x2 + 142x − 120
y3 − 7y + 6
x3 − 10x2 − 53x − 42
y3 − 2y2 − 29y − 42
2y3 − 5y2 − 19y + 42
x3 + 13x2 + 32x + 20
x3 − 3x2 − 9x − 5
2y3 + y2 − 2y − 1
x3 − 2x2 − x + 2
Factorize of the following polynomials:
x3 + 13x2 + 31x − 45 given that x + 9 is a factor
Factorize of the following polynomials:
4x3 + 20x2 + 33x + 18 given that 2x + 3 is a factor.
x4 − 2x3 − 7x2 + 8x + 12
x4 + 10x3 + 35x2 + 50x + 24
2x4 − 7x3 − 13x2 + 63x − 45
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.6 [Pages 33 - 34]
Define zero or root of a polynomial.
If \[x = \frac{1}{2}\] is a zero of the polynomial f(x) = 8x3 + ax2 − 4x + 2, find the value of a.
Write the remainder when the polynomialf(x) = x3 + x2 − 3x + 2 is divided by x + 1.
Find the remainder when x3 + 4x2 + 4x − 3 is divided by x.
If x + 1 is a factor of x3 + a, then write the value of a.
If f(x) = x4 − 2x3 + 3x2 − ax − b when divided by x − 1, the remainder is 6, then find the value of a + b
RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.7 [Pages 34 - 35]
Mark the correct alternative in each of the following:
If x − 2 is a factor of x2 + 3ax − 2a, then a =
2
-2
1
-1
If x3 + 6x2 + 4x + k is exactly divisible by x + 2, then k =
−6
−7
−8
−10
If x − a is a factor of x3 −3x2a + 2a2x + b, then the value of b is
0
2
1
3
If x140 + 2x151 + k is divisible by x + 1, then the value of k is
1
-3
2
-2
If x + 2 is a factor of x2 + mx + 14, then m =
7
2
9
14
If x − 3 is a factor of x2 − ax − 15, then a =
-2
5
-5
3
If x51 + 51 is divided by x + 1, the remainder is
0
1
49
50
If x + 1 is a factor of the polynomial 2x2 + kx, then k =
-2
-3
4
2
If x + a is a factor of x4 − a2x2 + 3x − 6a, then a =
0
-1
1
2
The value of k for which x − 1 is a factor of 4x3 + 3x2 − 4x + k, is
3
1
-2
-3
If x + 2 and x − 1 are the factors of x3 + 10x2 + mx + n, then the values of m and n are respectively
5 and −3
17 and −8
7 and −18
23 and −19
Let f(x) be a polynomial such that \[f\left( - \frac{1}{2} \right)\] = 0, then a factor of f(x) is
2x − 1
2x + 1
x − 1
x +1
When x3 − 2x2 + ax − b is divided by x2 − 2x − 3, the remainder is x − 6. The values of a and b are respectively
−2, −6
2 and −6
- 2 and 6
2 and 6
One factor of x4 + x2 − 20 is x2 + 5. The other factor is
x2 − 4
x − 4
x2 − 5
x + 2
If (x − 1) is a factor of polynomial f(x) but not of g(x) , then it must be a factor of
f(x) g(x)
−f(x) + g(x)
f(x) − g(x)
\[\left\{ f(x) + g(x) \right\} g(x)\]
(x+1) is a factor of xn + 1 only if
n is an odd integer
n is an even integer
n is a negative integer
n is a positive integer
If x2 + x + 1 is a factor of the polynomial 3x3 + 8x2 + 8x + 3 + 5k, then the value of k is
0
2/5
5/2
-1
If (3x − 1)7 = a7x7 + a6x6 + a5x5 +...+ a1x + a0, then a7 + a5 + ...+a1 + a0 =
0
1
128
64
If both x − 2 and \[x - \frac{1}{2}\] are factors of px2 + 5x + r, then
p = r
p + r = 0
2p + r = 0
p + 2r = 0
If x2 − 1 is a factor of ax4 + bx3 + cx2 + dx + e, then
a + c + e = b + d
a + b +e = c + d
a + b + c = d + e
b + c + d = a + e
Chapter 6: Factorisation of Polynomials
RD Sharma solutions for Mathematics for Class 9 chapter 6 - Factorisation of Polynomials
RD Sharma solutions for Mathematics for Class 9 chapter 6 (Factorisation of Polynomials) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 solutions in a manner that help students grasp basic concepts better and faster.
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Concepts covered in Mathematics for Class 9 chapter 6 Factorisation of Polynomials are Algebraic Identities, Polynomials, Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials, Factorising the Quadratic Polynomial (Trinomial) of the type ax2 + bx + c, a ≠ 0..
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