## RD Sharma Solutions Class 9 Chapter 6 Ex 6.4

**In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) , or not : (from 1 to 7)**

**Q1. f(x) = x ^{3} – 6x^{2} +11x – 6 , g(x) = x – 3**

Sol : Given here,

f(x) = x^{3} – 6x^{2} +11x – 6 andÂ g(x) = x â€“ 3

We have to prove, that g(x) = x -3 is the factor of f(x),

Therefore, we need to show, f(3) = 0

here , x â€“ 3 = 0

=> x = 3

Substitute the value of x = 3, in f(x), we get;

f(3) = 3^{3} â€“ 6*(3)^{2} +11(3) â€“ 6

= 27 â€“ (6*9) + 33 â€“ 6

= 27 â€“ 54 + 33 â€“ 6

= 60 â€“ 60

f(3) = 0

Therefore, g(x)=x -3 is the factor of f(x) =Â x^{3} – 6x^{2} +11x – 6.

**Q2. f(x) = 3x ^{4} + 17x^{3} + 9x^{2} – 7x – 10 , g(x) = x + 5**

Sol :

Given here , f(x) = 3x^{4} + 17x^{3} + 9x^{2} – 7x – 10

and g(x) = x + 5

To prove that g(x)=x + 5 is the factor of f(x) , we have to prove that;

f(-5) = 0

As, if we put, x + 5 = 0

=> x = -5

Now, Substitute the value of x=-5 in f(x)

f(-5) = 3(-5)^{4} + 17(-5)^{3}+ 9(-5)^{2} â€“ 7(-5) â€“ 10

= (3 * 625) + (12 * (-125)) +(9*25) + 35 â€“ 10

= 1875 â€“ 2125 + 225 + 35 â€“ 10

= 2135 â€“ 2135

f(-5) = 0

Therefore, g(x)=x + 5 is the factor of f(x) = 3x^{4} + 17x^{3} + 9x^{2} – 7x – 10.

**Q3. f(x) = x ^{5} + 3x^{4}– x^{3} – 3x^{2} + 5x + 15 , g(x)= x + 3.**

Ans:

Given here, f(x) = x^{5} + 3x^{4}– x^{3} – 3x^{2} + 5x + 15

and g(x) = x + 3

To prove that g(x) = x +3 is the factor of f(x) ,we have to prove that;

f(-3) = 0

As, x + 3 = 0

=> x = -3

Now, Substitute the value of x = -3, in f(x), we get;

f(-3) = (-3)^{5} + 3(-3)^{4} â€“ (-3)^{3} â€“ 3(-3)^{2} + 5(-3) + 15

= -243 + 243 + 27 â€“ 27 â€“ 15 + 15

f(-3)Â = 0

Therefore, g(x)=x + 3 is the factor of f(x) = x^{5} + 3x^{4}– x^{3} – 3x^{2} + 5x + 15.

**Q4. f(x) = x ^{3} – 6x^{2} – 19x + 84 , g(x) = x – 7**

Sol :

Given here,Â f(x) = x^{3} – 6x^{2}– 19x + 84

and g(x) = x â€“ 7

To prove that g(x) = x â€“ 7 is the factor of f(x) ,we have to prove that;

f(7) = 0

As, x â€“ 7 = 0

=> x = 7

Now, Substitute the value of x in f(x) to get;

f(7) = 7^{3} â€“ 6(7)^{2} â€“ 19(7) + 84

= 343 â€“ (6*49) â€“ (19*7) + 84

= 342 â€“ 294 â€“ 133 + 84

= 427 â€“ 427

f(7)Â = 0

Therefore, g(x)=x â€“ 7 is the factor of f(x) = x^{3} – 6x^{2}– 19x + 84.

**Q5. f(x) = 3x ^{3} + x^{2} – 20x + 12 , g(x) = 3x – 2**

Sol :

Given here, f(x) = 3x^{3} + x^{2} – 20x + 12

and g(x) = 3x â€“ 2

To prove that g(x)=3x-2 is the factor of f(x), we have to prove,

f(2/3) = 0

As, g(x) = 3x – 2 = 0

=> 3x = 2

=> x = 2/3

Now, Substitute the value of x=2/3 in f(x), to get;

f(2/3) = 3(2/3)^{3}Â + (2/3)^{3}â€“ 20(2/3) + 12

= 3(8/27) + 4/9 -40/3+ 12

= 8/9 + 4/9 – 40/3 + 12

= 12/9 – 40/3+ 12

Taking L.C.M of the denominators, 9, 3 and 1, we get

f(2/3) = \(\frac{12 – 120 + 108}{9}\)

= \(\frac{120 – 120 }{9}\)

f(2/3) = 0

Therefore, g(x)=3x-2 is the factor of f(x) = 3x^{3} + x^{2} – 20x + 12.

**Q6. \(f(x) = 2x^{3} – 9x^{2} + x + 13 , g(x) = 3 – 2x\)**

Sol :

Given here, \(f(x) = 2x^{3} – 9x^{2} + x + 13\)

g(x) = 3 â€“ 2x

To prove that g(x)=3-2x is the factor of f(x), we have to prove;

f(3/2) = 0

Since, 3 â€“ 2x = 0

=> -2x = -3

=> 2x = 3

=> x = 3/2

Now, Substitute the value of x=3/2 in f(x)

f(\(\frac{3}{2}\)) = 2(\(\frac{3}{2})^{3}\) â€“ 9(\(\frac{3}{2})^{2}\) + (\(\frac{3}{2}\)) + 13

= \(2(\frac{27}{8}) – 9(\frac{9}{4}) + \frac{3}{2} + 12\)

= \((\frac{27}{4}) – (\frac{81}{4}) + \frac{3}{2} + 12\)

Taking L.C.M of the denominators;

= \(\frac{21 – 81 + 6 + 48}{4}\)

= \(\frac{81 – 81}{4}\)

= 0

Therefore, g(x)=3-2x is the factor of f(x) = \(f(x) = 2x^{3} – 9x^{2} + x + 13\).

**Q7. \(f(x) = x^{3} – 6x^{2} + 11x – 6 , g(x) = x^{2} – 3x + 2\)**

Ans:

Given here,Â \(f(x) = x^{3} – 6x^{2} + 11x – 6\)

\(g(x) = x^{2} – 3x + 2\)

First we need to find the factors of g(x),Â \( x^{2} – 3x + 2\)

=>\( x^{2} – 2x â€“ x + 2\)

=> x(x â€“ 2) -1(x â€“ 2)

=> (x â€“ 1) and (x â€“ 2) are the factors of g(x)

To prove that g(x) is the factor of f(x), the results of f(1) and f(2) should be zero.

Let , x â€“ 1 = 0

x = 1

Now,substitute x=1 in f(x), we get;

\(f(1) = 1^{3} â€“ 6(1)^{2} + 11(1) â€“ 6\)

= 1 â€“ 6 + 11 â€“ 6

= 12 â€“ 12

= 0

Let , x â€“ 2 = 0

x = 2

Now, substitute x=2 in f(x), we get;

\(f(2) = 2^{3} â€“ 6(2)^{2} + 11(2) â€“ 6\)

= 8 â€“ (6 * 4) + 22 â€“ 6

= 8 â€“ 24 + 22 – 6

= 30 â€“ 30

= 0

Since, the results f(1) and f(2) are equal to 0. Therefore, g(x) is the factor of f(x)

**Q8. Show that (x â€“ 2) , (x + 3) and (x â€“ 4) are the factors of \(x^{3} – 3x^{2} – 10x + 24\)**

Sol :

Given here, f(x) = \(x^{3} – 3x^{2} – 10x + 24\)

The factors given are (x â€“ 2) , (x + 3) and (x â€“ 4)

To prove that g(x) is the factor of f(x) ,Â f(2) , f(-3) and f(4) should be equal to zero.

IfÂ x â€“ 2 = 0

then,Â x = 2

Now, Substitute x=2 in f(x)

f(2) = \(2^{3} â€“ 3(2)^{2} â€“ 10(2) + 24\)

= 8 â€“ (3 * 4) â€“ 20 + 24

= 8 â€“ 12 â€“ 20 + 24

= 32 â€“ 32

= 0

If x + 3 = 0

then, x = -3

Now, Substitute x = -3 in f(x)

f(-3) = \((-3)^{3} â€“ 3(-3)^{2} â€“ 10(-3) + 24\)

= -27 â€“ 3(9) + 30 + 24

= -27 â€“ 27 + 30 + 24

= 54 â€“ 54

= 0

If x – 4 = 0

then, x = 4

Now, SubstituteÂ x = 4 in f(x)

f(4) = \((4)^{3} â€“ 3(4)^{2} â€“ 10(4) + 24\)

= 64 â€“ (3*16) â€“ 40 + 24

= 64 â€“ 48 â€“ 40 + 24

= 84 â€“ 84

= 0

Since,Â f(2) , f(-3) and f(4) are equal to zero.Â Therefore, g(x) is the factor of f(x).

**Q9. Show that (x + 4) , (x â€“ 3) and (x â€“ 7) are the factors of \(x^{3} – 6x^{2} â€“ 19x + 84\)**

Sol :

Given here, f(x) = \(x^{3} – 6x^{2} – 19x + 84\)

The factors given here are (x + 4) , (x â€“ 3) and (x â€“ 7).

To prove that g(x) is the factor of f(x) , f(-4) , f(3) and f(7) should be equal to zero.

IfÂ x + 4 = 0, then,

=> x = -4

Now, substitute x = -4 in f(x)

f(-4) = \((-4)^{3} â€“ 6(-4)^{2} â€“ 19(-4) + 84\)

= -64 â€“ (6 * 16) â€“ ( 19 * (-4)) + 84

= -64 â€“ 96 + 76 + 84

= 160 â€“ 160

f(-4)= 0 ………1

If x â€“ 3 = 0, then,

=> x = 3

Now, substitute x = 3 in f(x)

f(3) = \((3)^{3} â€“ 6(3)^{2} â€“ 19(3) + 84\)

= 27 â€“ (6 * 9) â€“ ( 19 * 3) + 84

= 27 â€“ 54 â€“ 57 + 84

= 111 â€“ 111

f(3) = 0

If x â€“ 7 = 0, then.

=> x = 7

Now, substitute x = 7 in f(x)

f(7) = \((7)^{3} â€“ 6(7)^{2} â€“ 19(7) + 84\)

= 343 â€“ (6 * 49) â€“ ( 19 * 7) + 84

= 343 â€“ 294 – 133 + 84

= 427 â€“ 427

f(7) = 0

Since, f(-4) , f(3) and f(7) are equal to zero. therefore, g(x) is the factor of f(x).

**Q10. For what value of a is (x â€“ 5) a factor of \(x^{3} – 3x^{2} + ax – 10\)**

Sol :

Given here, f(x) = \(x^{3} – 3x^{2} + ax – 10\)

By factor theorem, we know,

If (x â€“ 5) is the factor of f(x) then , f(5) = 0

=> x â€“ 5 = 0

=> x = 5

Substitute x = 5 in f(x)

f(5) = \(5^{3} â€“ 3(5)^{2} + a(5) – 10\)

= 125 â€“ (3 * 25) + 5a â€“ 10

= 125 â€“ 75 + 5a â€“ 10

f(5) = 5a + 40

Equate f(5) to zero;

f(5) = 0

=> 5a + 40 = 0

=> 5a = -40

=> a = \(\frac{-40}{5}\)

a = -8

Therefore, when a= -8 , (x â€“ 5) will be factor of f(x).

**Q11. Find the value of a such that (x â€“ 4) is a factor of \(5x^{3} – 7x^{2} â€“ ax â€“ 28\)**

Sol :

Here, f(x) = \(5x^{3} – 7x^{2} â€“ ax â€“ 28\)

By factor theorem, we know,

If (x â€“ 4) is the factor of f(x) then , f(4) = 0

=> x â€“ 4 = 0

=> x = 4

Now, Substitute x = 4 in f(x), to get;

f(4) = \(5(4)^{3} â€“ 7(4)^{2} â€“ a(4) â€“ 28\)

= 5(64) â€“ 7(16) â€“ 4a â€“ 28

= 320 â€“ 112 â€“ 4a â€“ 28

f(4) = 180 â€“ 4

Equate f(4) to zero, to find the value of a;

f(4) = 0

=> 180 â€“ 4a = 0

=> -4a = -180

=> 4a = 180

=> a = 180/4

=> a = 45

Therefore, when a = 45 , (x â€“ 4) will be factor of f(x).

**Q12. Find the value of a, if (x + 2) is a factor of \(4x^{4} + 2x^{3} – 3x^{2} + 8x + 5a\)**

Sol :

Here, f(x) = \(4x^{4} + 2x^{3} – 3x^{2} + 8x + 5a\)

By factor theorem, we know;

If (x + 2) is the factor of f(x) then , f(-2) = 0

=> x + 2 = 0

=> x = -2

Now, substitute x = -2 in f(x), to get;

f(-2) = \(4(-2)^{4} + 2(-2)^{3} â€“ 3(-2)^{2} + 8(-2) + 5a\)

= 4(16) + 2(-8) â€“ 3( 4) â€“ 16 + 5a

= 64 â€“ 16 â€“ 12 â€“ 16 + 5a

= 5a + 20

Put f(-2) equal to zero

f(-2) = 0

=> 5a + 20 = 0

=> 5a = -20

=> a = -20/4

=> a = -4

Therefore, when a = -4 , (x + 2) will be factor of f(x).

**Q13. Find the value of k if x â€“ 3 is a factor of \(k^{2}x^{3} – kx^{2} + 3kx – k\).**

Sol :

Let f(x) = \(k^{2}x^{3} – kx^{2} + 3kx – k\)

From factor theorem, if x â€“ 3 is the factor of f(x) then f(3) = 0

=> x â€“ 3 = 0

=> x = 3

Substitute x = 3 in f(x), to get;

f(3) = \(k^{2}(3)^{3} â€“ k(3)^{2} + 3k(3) – k\)

= \(27k^{2} – 9k + 9k\) â€“ k

= \(27k^{2} â€“ k\)

f(3) = k(27k â€“ 1)

Equate f(3) to zero, to find the value of k;

=> f(3) = 0

=> k(27k â€“ 1) = 0

=> k = 0 and 27k â€“ 1 = 0

=> k = 0 and 27k = 1

=> k = 0 and k = 1/27

Therefore, when k = 0 and k=1/27 , (x â€“ 3) will be the factor of f(x).

**Q14. Find the values of a and b, if \(x^{2}\) â€“ 4 is a factor of \(ax^{4} + 2x^{3} – 3x^{2} + bx – 4\)**

Sol :

Given , f(x) = \(ax^{4} + 2x^{3} – 3x^{2} + bx – 4\)

g(x) = \(x^{2}\) â€“ 4

First we need to find the factors of g(x):

=> \(x^{2}\) â€“ 4

=> \(x^{2}\) = 4

=> x = \(\sqrt{4}\)

=> x = Â±2

Hence, (x â€“ 2) and (x + 2) are the factors of g(x).

By factor therorem, if (x â€“ 2) and (x + 2) are the factors of f(x) the result of f(2) and f(-2) should be zero.

Let , x â€“ 2 = 0

=> x = 2

Substitute the value of x=2 in f(x)

f(2) = \(a(2)^{4} + 2(2)^{3} â€“ 3(2)^{2} + b(2) – 4\)

= 16a + 2(8) â€“ 3(4) + 2b â€“ 4

= 16a + 2b + 16 â€“ 12 â€“ 4

f(2) = 16a + 2b

Equate f(2) equal to zero;

=> 16a + 2b = 0

=> 2(8a + b) = 0

=> 8a + b = 0Â Â Â Â Â Â Â Â ———- 1

Let , x + 2 = 0

=> x = -2

Substitute the value of x=-2 in f(x) to get;

f(-2) = \(a(-2)^{4} + 2(-2)^{3} â€“ 3(-2)^{2} + b(-2) – 4\)

= 16a + 2(-8) â€“ 3(4) – 2b â€“ 4

= 16a – 2b – 16 â€“ 12 â€“ 4

= 16a â€“ 2b â€“ 32

f(-2) = 16a – 2b â€“ 32

Equate f(2) equal to zero

=> 16a – 2b â€“ 32 = 0

=> 2(8a – b) = 32

=> 8a â€“ b = 16Â Â Â Â Â Â Â Â Â Â Â ———– 2

Solving equation 1 and 2, we get;

8a + b = 0

8a â€“ b = 16

16a = 16

a = 1

Now, substitute value of a in eq. 1, to get;

8(1) + b = 0

=> b = -8

Therefore, the values are a = 1 and b = -8.

**Q15. Find \(\alpha\; ,\; \beta\) if (x + 1) and (x + 2) are the factors of \(x^{3}+3x^{2}-2\alpha x+\beta\)**

Sol:

Given, f(x) = \(x^{3}+3x^{2}-2\alpha x+\beta\) and the factors are (x + 1) and (x + 2)

From factor theorem, if they are tha factors of f(x) then results of f(-2) and f(-1) should be zero

Let , x + 1 = 0

=> x = -1

Substitute value of x=-1 in f(x);

f(-1) = \((-1)^{3}+3(-1)^{2}-2\alpha (-1)+\beta\)

\(= -1 + 3 + 2\alpha + \beta\)

\(= 2\alpha + \beta\) + 2 ———— 1

Let , x + 2 = 0

=> x = -2

Substitute value of x=-2 in f(x);

f(-2) = \((-2)^{3}+3(-2)^{2}-2\alpha (-2)+\beta\)

\(= -8 + 12 + 4\alpha + \beta\)

\(= 4\alpha + \beta\) + 4 ————– 2

Solving 1 and 2 i.e ( 1 â€“ 2)

\(=> 2\alpha + \beta + 2 â€“ (4\alpha + \beta\) + 4 ) = 0

\(=> -2\alpha â€“ 2 = 0\)

\(=> 2\alpha = -2 \)

\(=> \alpha = -1\)

Substitute Î± = -1 in equation 1;

\(=> 2(-1) + \beta\) = -2

\(=> \beta \)= -2 + 2

=> Î² = 0

The values are Î±Â = -1 and Î² = 0.

**Q16. Find the values of p and q so that \(x^{4} + px^{3} + 2x^{2} -3x + q\) is divisible by (\(x^{2}\) â€“ 1)**

Sol :

Given here, f(x) = \(x^{4} + px^{3} + 2x^{2} -3x + q\)

g(x) = \(x^{2} – 1\)

First, we need to find the factors of \(x^{2} – 1\)

=> \(x^{2} – 1\) = 0

=> \(x^{2} = 1\)

=> x = Â±1

=> (x + 1) and (x – 1)

From factor theorem , if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0

Let us take , x + 1

=> x + 1 = 0

=> x = -1

Substitute the value of x in f(x)

f(-1) = \((-1)^{4} + p(-1)^{3} + 2(-1)^{2} -3(-1) + q\)

= 1 â€“ p + 2 + 3 + q

= -p + q + 6 ———- 1

Let us take , x – 1

=> x – 1 = 0

=> x = 1

Substitute the value of x in f(x)

f(1) = \((1)^{4} + p(1)^{3} + 2(1)^{2} -3(1) + q\)

= 1 + p + 2 â€“ 3 + q

= p + q ———- 2

Solve equations 1 and 2

-p + q = -6

p + q = 0

2q = -6

q = -3

substitute q value in equation 2

p + q = 0

p â€“ 3 = 0

p = 3

Therefore, the values of are p = 3 and q = -3

**Q17. Find the values of a and b so that (x + 1) and (x â€“ 1) are the factors of \(x^{4} + ax^{3} – 3x^{2} + 2x + b\).**

Sol :

Here, f(x) = \(x^{4} + ax^{3} – 3x^{2} + 2x + b\)

The factors are (x + 1) and (x â€“ 1)

From factor theorem , if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0

Let , us take x + 1

=> x + 1 = 0

=> x = -1

Substitute value of x in f(x), we get;

f(-1) = \((-1)^{4} + a(-1)^{3} â€“ 3(-1)^{2} + 2(-1) + b\)

= 1 â€“ a – 3 – 2 + b

= -a + b â€“ 4 ——- 1

Let , us take x â€“ 1

=> x â€“ 1 = 0

=> x = 1

Substitute value of x in f(x), we get;

f(1) = \((1)^{4} + a(1)^{3} â€“ 3(1)^{2} + 2(1) + b\)

= 1 + a – 3 + 2 + b

= a + b ——- 2

Solve equations 1 and 2

-a + b = 4

a + b = 0

2b = 4

b = 2

substitute value of b in eq 2, to get;

a + 2 = 0

a = -2

Therefore, the values are a = -2 and b = 2

**Q18. If \(x^{3} + ax^{2} – bx + 10\) is divisible by \(x^{3} – 3x + 2\), find the values of a and b**

Sol :

Here , f(x) = \(x^{3} + ax^{2} – bx + 10\)

g(x) = \(x^{3} – 3x + 2\)

first, we need to find the factors of g(x)

g(x) = \(x^{3} – 3x + 2\)

= \(x^{3} – 2x â€“ x + 2\)

= x(x â€“ 2) -1( x – 2)

= ( x â€“ 1) and ( x â€“ 2) are the factors

From factor theorem , if x = 1, 2 are the factors of f(x) then f(1) = 0 and f(2) = 0

Let, us take x â€“ 1

=> x â€“ 1 = 0

=> x = 1

Substitute the value of x in f(x)

f(1) = \(1^{3} + a(1)^{2} â€“ b(1) + 10\)

= 1 + a â€“ b + 10

= a â€“ b + 11 ——- 1

Let, us take x â€“ 2

=> x â€“ 2 = 0

=> x = 2

Substitute the value of x in f(x)

f(2) = \(2^{3} + a(2)^{2} â€“ b(2) + 10\)

= 8 + 4a â€“ 2b + 10

= 4a â€“ 2b + 18

Equate f(2) to zero

=> 4a â€“ 2b + 18 = 0

=> 2( 2a â€“ b + 9) = 0

=> 2a â€“ b + 9 ———- 2

Solve 1 and 2

a â€“ b = -11

2a â€“ b = -9

(-) (+) (+)

-a = -2

a = 2

substitute a value in eq 1

=> 2 â€“ b = -11

=> – b = -11 â€“ 2

=> -b = -13

=> b = 13

Therefore, the values are a = 2 and b = 13.

**Q19. If both ( x + 1) and (x â€“ 1) are the factors of \(ax^{3} + x^{2} -2x + b\) , Find the values of a and b**

Sol:

Here, f(x) = \(ax^{3} + x^{2} -2x + b\)

(x + 1) and (x â€“ 1) are the factors

From factor theorem , if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0

Let, x – 1= 0

=> x = -1

Substitute x value in f(x)

f(1) = \(a(1)^{3} + (1)^{2} -2(1) + b\)

= a + 1 â€“ 2 + b

= a + b â€“ 1 ———- 1

Let, x + 1= 0

=> x = -1

Substitute x value in f(x)

f(-1) = \(a(-1)^{3} + (-1)^{2} -2(-1) + b\)

= -a + 1 + 2 + b

= -a + b + 3 ———- 2

Solve equations 1 and 2

a + b = 1

-a + b = -3

2b = -2

=> b = -1

substitute b value in eq 1

=> a â€“ 1 = 1

=> a = 1 + 1

=> a = 2

Therefore, the values are a= 2 and b = -1

**Q20. What must be added to \(x^{3} – 3x^{2} – 12x + 19\) so that the result is exactly divisible by \(x^{2} + x – 6\).**

Sol :

Here , p(x) = \(x^{3} – 3x^{2} – 12x + 19\)

g(x) = \(x^{2} + x – 6\)

by division algorithm, when p(x) is divided by g(x) , the remainder wiil be a linear expression in x

let, r(x) = ax + b is added to p(x)

=> f(x) = p(x) + r(x)

= \(x^{3} – 3x^{2} – 12x + 19\) + ax + b

f(x) = \(x^{3} – 3x^{2} + x(a – 12) + 19\) + b

We know that , g(x) = \(x^{2} + x – 6\)

First, find the factors for g(x)

g(x) = \(x^{2} + 3x â€“ 2x – 6\)

= x(x + 3) -2(x + 3)

= (x + 3) ( x â€“ 2) are the factors

From, factor theorem when (x + 3) and (x â€“ 2) are the factors of f(x) the f(-3) = 0 and f(2) = 0

Let, x + 3 = 0

=> x = -3

Substitute the value of x in f(x)

f(-3) = \((-3)^{3} â€“ 3(-3)^{2} + (-3)(a – 12) + 19\) + b

= -27 â€“ 27 â€“ 3a + 24 + 19 + b

= -3a + b + 1 ——— 1

Let, x â€“ 2 = 0

=> x = 2

Substitute the value of x in f(x)

f(2) = \((2)^{3} â€“ 3(2)^{2} + (2)(a – 12) + 19\) + b

= 8 â€“ 12 + 2a â€“ 24 + b

= 2a + b â€“ 9 ——— 2

Solve equations 1 and 2

-3a + b = -1

2a + b = 9

(-) (-) (-)

-5a = – 10

a = 2

substitute the value of a in eq 1

=> -3(2) + b = -1

=> -6 + b = -1

=> b = -1 + 6

=> b = 5

\(âˆ´\) r(x) = ax + b

= 2x + 5

\(âˆ´\) \(x^{3} – 3x^{2} – 12x + 19\) is divided by \(x^{2} + x – 6\) when it is added by 2x + 5.

**Q21. What must be added to \(x^{3} – 6x^{2} – 15x + 80\) so that the result is exactly divisible by \(x^{2} + x â€“ 12\).**

Sol :

Let, p(x) = \(x^{3} – 6x^{2} – 15x + 80\)

q(x) = \(x^{2} + x â€“ 12\)

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 subtracted from p(x), so that p(x) â€“ q(x) is divisible by q(x)

let f(x) = p(x) â€“ q(x)

q(x) = \(x^{2} + x â€“ 12\)

= \(x^{2} + 4x – 3x â€“ 12\)

= x(x + 4) (-3)(x + 4)

= (x+4) , (x â€“ 3)

clearly, (x â€“ 3) and (x + 4) are factors of q(x)

so, f(x) wiil be divisible by q(x) if (x â€“ 3) and (x + 4) are factors of q(x)

from , factor theorem

f(-4) = 0 and f(3) = 0

=> f(3) = \(3^{3} â€“ 6(3)^{2} â€“ 3(a+15) + 80\) – b = 0

= 27 â€“ 54 -3a -45 + 80 â€“b

= -3a â€“b + 8 ——— 1

Similarly,

f(-4) = 0

=> f(-4) => \((-4)^{3} â€“ 6(-4)^{2} â€“ (-4)(a+15) + 80\) – b = 0

=> -64 â€“ 96 -4a + 60 + 80 â€“b = 0

=> 4a â€“ b â€“ 20 = 0 ———- 2

Substract eq 1 and 2

=> 4a â€“ b â€“ 20 â€“ 8 + 3a + b = 0

=> 7a â€“ 28 = 0

=> a = \(\frac{28}{7}\)

=> a= 4

Put a = 4 in eq 1

=> -3(4) â€“ b = -8

=> -b â€“ 12 = -8

=> -b = -8 + 12

=> b = -4

Substitute a and b values in r(x)

=> r(x) = ax + b

= 4x â€“ 4

Hence, p(x) is divisible by q(x) , if r(x) = 4x â€“ 4 is subtracted from it

**Q22. What must be added to \(3x^{3} + x^{2} – 22x + 9\) so that the result is exactly divisible by \(3x^{2} + 7x – 6\)**

Sol :

Let, p(x) = \(3x^{3} + x^{2} – 22x + 9\) and q(x) = \(3x^{2} + 7x – 6\)

By division theorem, when p(x) is divided by q(x) , the remainder is a linear equation in x.

Let, r(x) = ax + b is added to p(x) , so that p(x) + r(x) is divisible by q(x)

f(x) = p(x) + r(x)

=> f(x) = \(3x^{3} + x^{2} – 22x + 9(ax + b)\)

=> = \(3x^{3} + x^{2} + x(a â€“ 22) + b + 9\)

We know that,

q(x) = \(3x^{2} + 7x – 6\)

= \(3x^{2} + 9x â€“ 2x – 6\)

= 3x(x+3) â€“ 2(x+3)

= (3x-2) (x+3)

So, f(x) is divided by q(x) if (3x-2) and (x+3) are the factors of f(x)

From, factor theorem,

f(\(\frac{2}{3}\)) = 0 and f(-3) = 0

let , 3x â€“ 2 = 0

3x = 2

x = \(\frac{2}{3}\)

=> f(\(\frac{2}{3}\)) = \(3(\frac{2}{3})^{3} + (\frac{2}{3}\))^{2} + (\(\frac{2}{3}\))(a â€“ 22) + b + 9

= \(3(\frac{8}{27})+\frac{4}{9}+\frac{2}{3}a-\frac{44}{3}+b+9\)

=\(\frac{12}{9}+\frac{2}{3}a-\frac{44}{3}+b+9\)

= \(\frac{12+6a-132+9b+81}{9}\)

Equating to zero, we get;

=> \(\frac{12+6a-132+9b+81}{9}\) = 0

=> 6a + 9b â€“ 39 = 0

=> 3(2a + 3b â€“ 13) = 0

=> 2a + 3b â€“ 13 = 0 ———- 1

Similarly,

say, x + 3 = 0

=> x = -3

=> f(-3) = \(3(-3)^{3} + (-3)^{2} + (-3)(a â€“ 22) + b + 9\)

= -81 + 9 -3a + 66 + b + 9

= -3a + b + 3

Equating to zero, we get;

-3a + b + 3 = 0

Multiply by 3

-9a + 3b + 9 = 0 ——– 2

Minus eq 1 from eq.2;

=> -9a + 3b + 9 -2a â€“ 3b + 13 = 0

=> -11a + 22 = 0

=> -11a = -22

=> a = \(\frac{22}{11}\)

=> a = 2

Put value of a in eq 1

=> -3(2) + b = -3

=> -6 + b = -3

=> b = -3 + 6

=> b = 3

Put the values in r(x)

r(x) = ax + b

= 2x + 3

Hence, p(x) is divisible by q(x) , if r(x) = 2x + 3 is added to it.

**Q23. If x â€“ 2 is a factor of each of the following two polynomials , find the value of a in each case :**

**\(x^{3} – 2ax^{2} + ax – 1\)****\(x^{5} – 3x^{4} – ax^{3} + 3ax^{2} + 2ax + 4\)**

Sol :

(i) Assume f(x) = \(x^{3} – 2ax^{2} + ax â€“ 1\)

Using factor theorem, we know;

if (x â€“ 2) is the factor of f(x) the f(2) = 0

Assume,

x â€“ 2 = 0

=> x = 2

Put x=2 value in f(x);

f(2) = \(2^{3} – 2a(2)^{2} + a(2) â€“ 1\)

= 8 â€“ 8a + 2a â€“ 1

= -6a + 7

Substitute f(2) equal to zero;

=> -6a + 7 = 0

=> -6a = -7

=> a= 7/6

Therefore, when (x – 2 ) is the factor of f(x) then a= \(\frac{7}{6}\)

(ii) Assume f(x) = \(x^{5} – 3x^{4} – ax^{3} + 3ax^{2} + 2ax + 4\)

Using factor theorem,

if (x â€“ 2) is the factor of f(x) the f(2) = 0

If x â€“ 2 = 0

=> x = 2

Put x value in f(x)

f(2) = \(2^{5} â€“ 3(2)^{4} â€“ a(2)^{3} + 3a(2)^{2} + 2a(2) + 4\)

= 32 â€“ 48 â€“ 8a + 12 + 4a + 4

= 8a â€“ 12

Put f(2) equal to zero

=> 8a â€“ 12 = 0

=> 8a = 12

=> a = \(\frac{12}{8}\)

= \(\frac{3}{2}\)

Therefore, when (x â€“ 2) is a factor of f(x) then a= \(\frac{3}{2}\).

**Q24. In each of the following two polynomials , find the value of a, if (x â€“ a) is a factor :**

**\(x^{6} – ax^{5} + x^{4} – ax^{3} + 3x – a + 2\)****\(x^{5} – a^{2}x^{3} + 2x + a + 1\)**

Sol :

(i) \(x^{6} – ax^{5} + x^{4} – ax^{3} + 3x – a + 2\)

Say, f(x) = \(x^{6} – ax^{5} + x^{4} – ax^{3} + 3x – a + 2\)

As, x â€“ a = 0

=> x = a

Put the value of x in f(x)

f(a) = \(a^{6} â€“ a(a)^{5} + (a)^{4} â€“ a(a)^{3} + 3(a) – a + 2\)

= \(a^{6} â€“ a^{6} + (a)^{4} â€“ a^{4} + 3(a) – a + 2\)

= 2a + 2

Put f(a) equal to zero

=> 2a + 2 = 0

=> 2(a + 1) = 0

=> a = -1

Thus, when (x â€“ a) is a factor of f(x) then a = -1

(ii) \(x^{5} – a^{2}x^{3} + 2x + a + 1\)

let, \(f(x) = x^{5} – a^{2}x^{3} + 2x + a + 1\)

here , x â€“ a = 0

=> x = a

Substitute the value of x in f(x)

f(a) = \(a^{5} – a^{2}a^{3} + 2(a) + a + 1\)

= \(a^{5} – a^{5} + 2a + a + 1\)

= 3a + 1

Put f(a) equal to zero

=> 3a + 1 = 0

=> 3a = -1

=> a= \(\frac{-1}{3}\)

So, when (x â€“ a) is a factor of f(x) then a = \(\frac{-1}{3}\)

**Q25. In each of the following two polynomials , find the value of a, if (x + a) is a factor :**

**\(x^{3} + ax^{2} – 2x + a + 4\)****\(x^{4} – a^{2}x^{2} + 3x – a\)**

Sol :

(i) \(x^{3} + ax^{2} – 2x + a + 4\)

say, f(x) = \(x^{3} + ax^{2} – 2x + a + 4\)

As, x + a = 0

=> x = -a

Put the value of x in f(x)

f(-a) = \((-a)^{3} + a(-a)^{2} â€“ 2(-a) + a + 4\)

= \((-a)^{3} + a^{3} â€“ 2(-a) + a + 4\)

= 3a + 4

Equate f(-a) to zero

=> 3a + 4 = 0

=> 3a = -4

=> a = \(\frac{-4}{3}\)

Therefore, when (x + a) is a factor of f(x) then a = \(\frac{-4}{3}\)

(ii) \(x^{4} – a^{2}x^{2} + 3x – a\)

Say, \(f(x) = x^{4} – a^{2}x^{2} + 3x – a\)

As , x + a = 0

=> x = -a

Put the value of x in f(x)

f(-a) = \((-a)^{4} – a^{2}(-a)^{2} + 3(-a) – a\)

= \(a^{4} – a^{4} – 3(a) – a\)

= -4a

Put f(-a)Â equal to zero

=> -4a = 0

=> a = 0

Therefore, when (x + a) is a factor of f(x) then a = 0