Chapter 6 Factorisation of Algebraic Expressions RD Sharma Solutions Exercise 6.4 Class 9 Maths
Chapter Name  RD Sharma Chapter 6 Factorisation of Polynomials Exercise 6.4 
Book Name  RD Sharma Mathematics for Class 10 
Other Exercises 

Related Study  NCERT Solutions for Class 10 Maths 
Exercise 6.4 Solutions
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)
1. f(x) = x^{3}  6x^{2} + 11x  6; g(x) = x  3
Solution
We have f(x) = x^{3}  6x^{2} + 11x  6 and g(x) = x  3
In order to find whether polynomial g(x) = x  3 is a factor of f(x), it is sufficient to show that f(3) =0
Solution
We have
f(x) = 3x^{3} + 17x^{3} + 9x^{3}  7x  10; and g(x) = x + 5
In order to find whether g(x) = x  (5) is a factor of f(x) or not, it is sufficient to show that f(5) = 0
3. f(x) = x^{5} + 3x^{4}  x^{3}  3x^{2} + 5x + 15, g(x) = x + 3
Solution
We have
f(x) = x^{5} + 3x^{4}  x^{3}  3x^{2} + 5x + 15, and g(x) = x + 3
In order to find whether g(x) = x (3) is a factor of f(x) or not, it is sufficient to prove that f(3) = 0
4. f(x) = x^{3}  6x^{3}  19x + 84, g(x) = x  7
Solution
We have, f(x) = x^{3}  6x^{3}  19x + 84, and g(x) = x  7
In order to find whether g(x) = x 7 is a factor of f(x) or not, it is sufficient to show that f(7) = 0
In order to find whether g(x) = x 7 is a factor of f(x) or not, it is sufficient to show that f(7) = 0
5. f(x) = 3x^{3} + x^{2}  20x + 12 and g(x) = 3x  2
Solution
We have
f(x) = 3x^{3} + x^{2}  20x + 12 and g(x) = 3x  2
In order to find whether g(x) = 3(x  2/3) is a factor of f(x) or not, it is sufficient to prove
that f(2/3) = 0
6. f(x) = 2x^{3}  9x^{2} + x + 12, g(x) = 3  2x
Solution
We have f(x) = 2x^{3}  9x^{2} + x + 12, and g(x) = 3  2x
In order to find whether g(x) = 3  2x = 2(x  3/2) is a factor of f(x) or not, it is sufficient to prove that f(3/2) = 0
7. f(x) = x^{3}  6x^{2} + 11x  6, g(x) = x^{3}  3x + 2
Solution
We have
f(x) = x^{3}  6x^{2} + 11x  6, g(x) = x^{3}  3x + 2
f(x) = x^{3}  6x^{2} + 11x  6, g(x) = x^{3}  3x + 2
⇒ g(x) = x^{2}  3x + 2 = (x  1)(x  2)
Clearly, (x  1) and (x  2) are factors of g(x)
In order to find whether g(x) = (x  1)(x  2) is a factor of f(x) or not, it is sufficient to prove that (x  1) and (x  2) are factors of f(x).
Clearly, (x  1) and (x  2) are factors of g(x)
In order to find whether g(x) = (x  1)(x  2) is a factor of f(x) or not, it is sufficient to prove that (x  1) and (x  2) are factors of f(x).
i.e., we should prove that f(1) = 0 and f(2) = 0
8. Show that (x  2), (x + 3) and (x  4) are factors of x^{3}  3x^{2}  10x + 24.
Solution
Let f(x) = x^{3}  3x^{2}  10x + 24 be the given polynomial.
In order to prove that (x  2), (x + 3), (x  4) are factors of f(x), it is sufficient to prove that f(2) = 0, f(3) = 0 and f(4) = 0 respectively.
In order to prove that (x  2), (x + 3), (x  4) are factors of f(x), it is sufficient to prove that f(2) = 0, f(3) = 0 and f(4) = 0 respectively.
Solution
Let f(x) = x^{3}  6x^{2}  19 + 84 be the given polynomial
In order to prove that (x + 4), (x  3) and (x  7) are factors of f(x), it is sufficient to prove that f(4) = 0 , f(3) = 0 and f(7) = 0 respectively
In order to prove that (x + 4), (x  3) and (x  7) are factors of f(x), it is sufficient to prove that f(4) = 0 , f(3) = 0 and f(7) = 0 respectively
10. For what value of a is (x  5) a factor of x^{3}  3x^{3} + ax  10 ?
Solution
Let f(x) = x^{3}  3x^{3} + ax  10 be the given polynomial
From factor theorem,
If (x  5) is a factor of f(x) then f(5) = 0
From factor theorem,
If (x  5) is a factor of f(x) then f(5) = 0
11. Find the value of a such that (x  4) is a factor of 5x^{3}  7x^{2}  ax  28.
Solution
Let f(x) = 5x^{3}  7x^{2}  ax  28 be the given polynomial from factor theorem, if (x  4) is a factor of f(x) then f(4) = 0
Hence (x  4) is a factor of f(x) when a = 45
12. For what value of a, if x + 2 is a factor of factor of 4x^{4} + 2x^{3}  3x^{2} + 8x + 5a.
Solution
Let f(x) = 4x^{4} + 2x^{3}  3x^{2} + 8x + 5a be the given polynomial
From factor theorem if (x + 2) is a factor of f(x) then f(2) = 0
From factor theorem if (x + 2) is a factor of f(x) then f(2) = 0
Solution
Let f(x) = k^{2}x^{3}  kx^{2} + 3kx  k be the given polynomial from factor theorem if (x  3) is a factor of f(x) then f(3) = 0
14. Find the values of a and b, if x^{2}  4 is a factor of ax^{4} + 2x^{3}  3x^{2} + bx  4.
Solution
Let f(x) = ax^{4} + 2x^{3}  3x^{2} + bx  4 and g(x) = x^{2}  4
We have g(x) = x^{2}  4 = (x  2)(x + 2)
Given g(x) is a factor of f(x).
We have g(x) = x^{2}  4 = (x  2)(x + 2)
Given g(x) is a factor of f(x).
15. Find Î± and Î², if x + 1 and x + 2 are factors of x^{3} + 3x^{2}  2ax +Î².
Solution
Let f(x) = x^{3} + 3x^{2}  2ax +Î² be the given polynomial from factor theorem, if (x + 1) and (x + 2) are factors of f(x) then f(1) = 0 and f(2) = 0
16. Find the values of p and q so that x^{3} + px^{3}  2x^{2}  3x + q is divisible by (x^{2}  1)
Solution
Let f(x) = x^{3} + px^{3} + 2x^{3}  3x + q be the given polynomial and let g(x) = x^{3}  1 = (x  1)(x + 1)
Clearly, (x  1) and (x + 1) are factors of g(x)
Given g(x) is a factor of f(x)
⇒ (x  1) and (x + 1) are factors of f(x)
From factor theorem,
If (x  1) and (X + 1) are factors of f(x) then f(1) = 0 and f1) = 0 respectively
Clearly, (x  1) and (x + 1) are factors of g(x)
Given g(x) is a factor of f(x)
⇒ (x  1) and (x + 1) are factors of f(x)
From factor theorem,
If (x  1) and (X + 1) are factors of f(x) then f(1) = 0 and f1) = 0 respectively
17. Find the values of a and b so that (x + 1) and (x  1) are factors of x^{4} + ax^{3}  3x^{2} + 2x + b.
Solution
Let f(x) = x^{4} + ax^{3}  3x^{2} + 2x + b be the given polynomial.
From factor theorem ; if (x + 1) and (x  1) are factors of f(x) then f(1) = 0 and f(1) = 0 respectively.
18. If x^{3} + ax^{2}  bx + 10 is divisible by x^{2}  3x + 2, find the values of a and b.
Solution
Let f(x) = x^{3} + ax^{2}  bx + 10 and g(x) = x^{2}  3x + 2 be the given polynomials.
We have g(x) = x^{2}  3x + 2 = (x  2)(x  1)
⇒Clearly, (x  1) and (x  2) are factors of g(x)
Given that f(x), is divisible by g(x)
⇒ g(x) is a factor of f(x)
⇒ (x  2) and (x  1) are factors of f(x)
From factor theorem,
If (x  1) and (x  2) are factors of f(x) then f(1) = 0 and f(2) = 0 respectively.
Given that f(x), is divisible by g(x)
⇒ g(x) is a factor of f(x)
⇒ (x  2) and (x  1) are factors of f(x)
From factor theorem,
If (x  1) and (x  2) are factors of f(x) then f(1) = 0 and f(2) = 0 respectively.
19. If both x + 1 and x  1 are factors of ax^{3} + x^{2}  2x + b, find the values of a and b.
Solution
Let f(x) = ax^{3} + x^{2}  2x + b be the given polynomial.
Given (x + 1) and (x  1) are factor of f(x).
From factor theorem,
If (x + 1) and (x  1) are factors of f(x) then f(1) = 0 and f(1) = 0 respectively.
From factor theorem,
If (x + 1) and (x  1) are factors of f(x) then f(1) = 0 and f(1) = 0 respectively.
20. What must be added to x^{3}  3x^{2}  12x + 19 so that the result is exactly divisible by x^{2} + x  6 ?
Solution
Let p(x) = x^{3}  3x^{2}  12x + 19 and q(x) = x^{2} + x  6.
By division algorithm, when p(x) is divided by q(x), the remainder is a linear expression in x.
By division algorithm, when p(x) is divided by q(x), the remainder is a linear expression in x.
So, let r(x) = ax + b is added to p(x) so that p(x) + r(x) is divisible by q(x).
Let f(x) = p(x)+ r(x)
21. What must be subtracted from x^{3}  6x^{2}  15x + 80 so that the result is exactly divisible by x^{2} + x  12 ?
Solution
Let p(x) = x^{3}  6x^{2}  15x + 80 and q(x) = x^{2} + x  12
By division algorithm, when p(x) is divided nu q(x) the remainder is a linear expression in x.
So, let r(x) = ax + b is subtracted from p(x), So that p(x)  r(x) is divisible by q(x)
Let f(x)  p(x)  r(x)
Clearly, (3x  2) and (x + 3) are factors of q(x)
Therefore, f(x) will be divisible by q(x) if (3x  2) and (x + 3) are factors of f(x) i.e., from factor theorem,
By division algorithm, when p(x) is divided nu q(x) the remainder is a linear expression in x.
So, let r(x) = ax + b is subtracted from p(x), So that p(x)  r(x) is divisible by q(x)
Let f(x)  p(x)  r(x)
Clearly, (3x  2) and (x + 3) are factors of q(x)
Therefore, f(x) will be divisible by q(x) if (3x  2) and (x + 3) are factors of f(x) i.e., from factor theorem,
22. What must be added to 3x^{3} + x^{2}  22x + 9 so that the result is exactly divisible by 3x^{3} + 7x  6?
Solution
Let p(x) = 3x^{3} + x^{2}  22x + 9 and q(x) = 3x^{3} + 7x  6
By division algorithm,
When p(x) is divided by q(x), the remainder is a linear equation in x.
So, let r(x) = ax + b is added to p(x), so that p(x) + r(x) is divisible by q(x)
Let f(x) = p(x) + r(x)
When p(x) is divided by q(x), the remainder is a linear equation in x.
So, let r(x) = ax + b is added to p(x), so that p(x) + r(x) is divisible by q(x)
Let f(x) = p(x) + r(x)
23. If x  2 is a factor of each of the following two polynomials, find the values of a in each case :
(i) x^{3}  2ax^{2} + ax  1
(ii) x^{5}  3x^{4}  ax^{3} + 3ax^{2} + 2ax + 4
(ii) x^{5}  3x^{4}  ax^{3} + 3ax^{2} + 2ax + 4
Solution
24. In each of the following two polynomials, find the value of a, if x  a is a factor:
(i) x^{6}  ax^{5} + x^{4}  ax^{3} + 3x  a + 2
(ii) x^{5}  a^{2}x^{3} + 2x + a + 1
Solution(ii) x^{5}  a^{2}x^{3} + 2x + a + 1
Let f(x) = x^{6}  ax^{5} + x^{4}  ax^{3} + 3x  a + 2 be the given polynomial.
From factor theorem,
If (x  a) is a factor of f(x) then f(a) = 0 [ ∵ x  a = 0 ⇒ x = a]
25. In each of the following two polynomials, find the value of a, if x + a is a factor.
(i) x^{3} + ax^{2}  2x + a + 4
(ii) x^{4}  a^{2}x^{2} + 3x  a
(i) x^{3} + ax^{2}  2x + a + 4
(ii) x^{4}  a^{2}x^{2} + 3x  a
Solution