KC Sinha Mathematics Solution Class 9 Chapter 3 Polynomials exercise 3.2

Page 3.19

Exercise 3.2

Question 1

Fill up the blanks :
(i) When x²+2x-3 is divided by (x-1) , remainder = __
Sol :
Divisor=x-1
Zero of x-1 is 1
∴ By remainder theorem ,
Remainder=p(1)=(1)²+2(1)-3
=1+2-3
=0

(ii) If (x-2) is a factor of the polynomial p(x) , then p(2)= __
Sol :
⇒By factor theorem : Let p(x) be a polynomial in x of degree n≥1 and let a be any real number , then (x-a) is a factor of p(x) if and only if p(a)=0
⇒It is already given that (x-2) is factor of p(x) then according to theorem , p(2)=0

Question 2

What will be the remainder when y⁴-3y²+2y+1 is divided by (y-1) ?
Sol :
Divisor=y-1
Zero of y-1 is 1
∴ By remainder theorem ,
Remainder=p(1)=(1)⁴-3(1)²+2(1)+1
=1-3+2+1
=4-3
=1

Question 3

What will be the remainder when x²+4x+2 is divided by (x+2) ?
Sol :
Divisor=x+2
Zero of x+2 is -2
∴ By remainder theorem ,
Remainder=p(-2)=(-2)²+4(-2)+2
=4-8+2
=-2

Question 4

If for polynomial p(x) , p(-1)=3 , then what will be the remainder when p(x) is divided by (x+1) ?
Sol :
x+1=0
x=-1

=3

Question 5

If for polynomial p(x) , $p\left(-\dfrac{2}{3}\right)=0$ , then write a factor of polynomial p(x)
Sol :
$x=-\frac{2}{3}$
3x=-2
3x+2=0

3x+2

Question 6

If for polynomial p(x), p(-3)=0 , then write a factor of polynomial p(x).
Sol :
x=-3 , p(-3)=2

p(x)-2=p(-3)-2
=2-2
=0

x=-3
x+3=0

x+3

Question 7

Find the remainder when x³ + 3x² + 3x + 1 is divided by
(i) x+1
Sol :
p(x)=x³+3x²+3x+1
x+1=0
x=-1

p(-1)=(-1)³+3(-1)²+3(1)+1

=-1+3-3+1

(ii) $x-\dfrac{1}{2}$
Sol :
$x-\frac{1}{2}=0$

$x=\frac{1}{2}$

$p\left(\frac{1}{2}\right)=\left(\frac{1}{2}\right)^3+3\left(\frac{1}{2}\right)^2+3\left(\frac{1}{2}\right)+1$

$=\frac{1}{8}+\frac{3}{4}+\frac{3}{2}+1$

$=\frac{1+6+12+8}{8}=\frac{27}{8}$

Question 8

(i) Find the remainder when x³+1 is divided by x + 1
Sol :
0
(ii) Find the remainder when x⁴+x³-2x²+x+1 is divided by x-1
Sol :
2
(iii) Find the remainder when x3-ax²+6x-a is divided by x - a.
Sol :

5a

Question 9

If p(x) = x⁴ - 3x³ + 2x + 1 , then using remainder theorem, f‌ind the remainder when p(x) is divided by :
(i) x-2
Sol :
9
(ii) x-4
Sol :
217

Question 10

If p(x)=4x³-3x²+2x-4 , then using remainder theorem, f‌ind the remainder when p(x) is divided by :
(i) x-1
Sol :
-1
(ii) x+1
Sol :

-13

Question 11

If p(x)=x²+4x+2 , then what will be the remainder when p(x) is divided by x+2 ?
Sol :
-2

Question 12

Using factor theorem determine whether x-1 is a factor of the following:
(i) x³+x²-2x+1
Sol :
x-1=0
x=1
p(x)=x³+x²-2x+1
p(1)=1³+1²-2(1)+1

=1+1-2+1

No

(ii) 8x⁴-12x³+18x+14
Sol :

No

(iii) x³+8x²-7x-2
Sol :

Yes

(iv) $2\sqrt{2}x^3+5\sqrt{2}x^2-7\sqrt{2}$
Sol :
x-1=0
x=1

p(x)=$2\sqrt{2}x^3+5\sqrt{2}x^2-7\sqrt{2}$

p(1)=$2\sqrt{2}(1)^3+5\sqrt{2}(1)^2-7\sqrt{2}$

=2√2+5√2-7√2

Yes

Question 13

Use factor theorem to determine whether g(x) is a factor of p(x) in each of the following cases :
(i) p(x)=2x³+x²-2x-1 , g(x)=x+1
Sol :
x+1=0
x=-1

p(-1)=2(-1)³+(-1)²-2(-1)-1
=-2+1+2-1
=0

Yes

(ii) p(x)=x³+3x²+3x+1 , g(x)=x+2
Sol :

No

(iii) p(x)=x³-4x²+x+6 , g(x)=x-3
Sol :

Yes

Question 14

(i) Examine whether 7 + 3x is a factor of 3x³ + 7x.
Sol :
7+3x=0
3x=-7
$x=\frac{-7}{3}$

p(x)=3x³+7x
$p\left(\frac{-7}{3}\right)=3\left(\frac{-7}{3}\right)^3+7\left(\frac{-7}{3}\right)$
$=3\left(\frac{-343}{7}\right)-\frac{49}{3}$
$=\frac{-343}{9}-\frac{49}{3}$

No

(ii) Examine whether x+2 is a factor of the polynomials x³+3x²+5x+6 and 2x+4
Sol :
x+2=0
x=-2

p(x)=x³+3x²+5x+6 , g(x)=2x+4
p(-2)=(-2)³+3(-2)²+5(-2)+6
=-8+12-10+6
=-18+18
=0

x+2, p(x) का गुणनखण्ड है ।

g(-2)=2(-2)+4
=-4+4
=0

Yes

(iii) Examine whether q(t)=4t³+4t²-t-1 is a multiple of 2t+1
Sol :

2t+1

2t=-1

$t=-\frac{1}{2}$

$q\left(-\frac{1}{2}\right)=4\left(-\frac{1}{2}\right)^2-\left(-\frac{1}{2}\right)-1$
$=4\left(-\frac{1}{8}\right)+4\left(\frac{1}{4}\right)+\frac{1}{2}+1$
$=-\frac{1}{2}+1+\frac{1}{2}-1$
=0

Yes

Question 15

Using factor theorem, determine whether g(x) is a factor of p(x) in the following pair of polynomials :
(i) p(x)=x³-3x²+2x-12 and g(x)=x-2
Sol :

No

(ii) p(x)=x³+x²+3x+175 and g(x)=x+5
Sol :

No

(iii) p(x)=2x³+4x²-5x-10 and g(x)=x+2
Sol :

Yes

(iv) $p(x)=2\sqrt{2}x^2+5x+\sqrt{2}$ and g(x)=x+√2
Sol :
g(x)=x+√2
x+√2=0
x=-√2

$p(x)=2\sqrt{2}x^2+5x+\sqrt{2}$
$=2\sqrt{2}(-\sqrt{2})^2+5(-\sqrt{2})+\sqrt{2}$
=4√2-5√2+√2
=5√2-5√2
=0

Yes

Question 16

Using factor theorem, show that x³-6x²+11x-6 is divided by x-1
Sol :

Question 17

(a) In the following polynomials if x- 2 is a factor of each polynomial, then f‌ind the value of a in each case :
(i) x²-3x+5a
Sol :
2/5

(ii) x³-2ax²+ax-1
Sol :
7/6

(b) If x-1 is a factor of polynomial ax³-4ax+4a-1 , find the value of a
Sol :
1

(c) In the following polynomial if x+a is a factor of each polynomial , find the value of a in each case.
(i) x³+ax²-2x+a+4
Sol :
-4/3

(ii) x4-a²x²+3x-a
Sol :

0

Question 18

If p(x)=x³+kx²+hx+6 and x+1 and x-2 are factors of p(x) , then find the value of h and k
Sol :
h=1 , k=-4

Question 19

If p(x)=x⁴-5x³+4x²+ax+b and x-1 and x-2 are factors of p(x) , find the values of a and b
Sol :
a=8, b=-8

Question 20

If x-1 and x+3 are factors of polynomial f(x)=x²-hx²-13x+k , find the values of h and k
Sol :
h=3 , k=15

Question 21

If (x-1) and (x-4) are factors of polynomials p(x)=(x²-3x+2)(x²+7x+a) and q(x)=(x²+5x+4)(x²-5x+b) , find the values of a and b
Sol :
a=12 , b=4

Question 22

If f(x)=x²+px+q , g(x)=x²+lx+m and each is divisible by x+a , then prove that $a=\dfrac{m-q}{l-p}$
Sol :