Zero is a rational or an irrational number?

find zeros of (x+2) (x-3)

solve for x and y a2/x-b2/y=0,a2b/x+b2a/y=a+b where x and y are not equal to zero

## Is zero (0) a rational number ? justify your answer ?

Yes, zero is a rational number

zero can be expressed as      etc

which are in the form of,   where p and q are integers and q  0.

Not defined

Binomial

## What is the zero of the zero polynomial ?

Every real number is a zero of the zero polynomial

## Write the number of zeroes in a cubic polynomial

No of the zeroes of cubic polynomial  = 3

## If -4 is a zero of the polynomial      then calculate the value of k.

Given ,

Since -4 is a zero of polynomial

## Write the zeroes of the polynomial

For zeroes , put

## if y = 2  and  y = 0  are the zeroes of the polynomial    find the value of a and b

.......................(1)

## Find the remainder when     is divided by x - a

Here ,

and the zero of x - a is a.

so,

So, by the remainder theorem 5a is the remainder when     is divided by x-a

## Using remainder theorem, factorize

Factors of

is a zero of    is a factor

## Factorize :

factor of  6 =

is zero of p (x) of (x+1) is a factor of p (x)

## factorize

(x+2)  is a factor of

Factors are: (x+2)   (x+1)  (x+10)

Alternative method

Factors  of  20 =

is a zero of p(x), and (x+1) is a factor of p(x)

Then

## Verify if 1 and  - 3  are zeroes of the polynomial   if yes, factorize the polynomial.

is a zero of p(x)

-3 is a zero of p(x)

is a factor of p(x)

, when  we divide physically

Hence,

## Using factor theorem, find the value of a if  is divisible be 2x+1.

If  (2x+1)is a factor of p(x) then 2x+1= 0,

is a zero  of the polynomial   p(x)

so,

## With out actual division show that  is exactly divisible by

Let,

Zero of x -2 is 2 as  x-2=0   x = 2

Zero of x -1 is 1 as  x-1=0    x = 1

Now,

and (x-2) are the factors of f (x)

is exactly diviible by g (x)

=3

## Find the point at which the equation  meets the x -axis

On the x-axis, y coordinate is zero.

So, put y = 0 in

we get,

meets the x-axis at  (2,0)

## If zeroes  of the polynomial   are  and  , then find the value of a.

Products of roots (zeroes)

## Check whether 4n can be  end with the digit  0  for any natural number n.

If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5.That is, the prime factorization of 4n would contain the prime  5. This is not possible  because 4n = (2)2n ; so the only prime in the factorization of 4n is2. So, the uniqueness of the Fundamental Theorem of arithmetic guarantees that there are no other primes in the factorization of 4n. So there is no natural number  n for which 4n ends with the digit zero

## Show that 7n cannot end with the digit zero, for any natural number n.

7n =(1x 7)n = 1n x 7n

So the only prime in the factorization of  7n is  7, not 2 or  5.

7n cannot end will the digit zero.

## Check wether (15)n can end with digit 0 for any

(15)n can end with the digit 0 only if (15)n  is divisible by 2 and  5.

But prime factors of (15)n are 3n x 5n

By Fundamental theorem of arithmetic, there is no natural number n for which (15)n ends with the digit zero

## Find the HCF 256 and 36 using Euclid's division Alogaritham. Also find their LCM and verify that HCF and LCM = Product of two numbers.

Since 256 > 36, we state with 256 as dividend and 36 as divisor, we have

256 = 36 x 7 +4

Now, 4 is the remainde, which is not zero, so we again apply Euclid Division Alogaritham to  36 and 4 we have

36 = 4 x 9 +0

Now, remainder has become zero and 4 is the divisor.

Hence the HCF of 256 and 36 is 4

LCM

256 = 28

36 = 22 x 32

LCM  (36,256) = 28 x 32 = 256 x 9

= 2304

HCF x LCM  = Product of the two number

4 x 2, 304 = 256 x 36

9216 = 9,216

## Find the quadratic polynomial whose sum and product of the zeroes are  and   respectively.

According to the question,

Sum of zeroes  =

and product of zeroes  =

= (Sum of zeroes ) + product of zeroes

## If 'm' and 'n' are the zeroes of the polynomial   find the value of

Let,

So, zeroes are,    and

Now,

## If    and   are the zeroes  of a polynomial     then find the value of

If    and    are the zeroes  of

then

and

## If one of the zeroes of the quadratic polynomial is  is a negative of the other, find the value of 'k'.

Given

Let one zero  be ,

The other  =

Sum of zeroes =

Sum of zeroes  =

According to the equation,

Sum of zeroes  =

## If the zeroes of the polynomial    are double in value to the zeroes of  , find the value of  p and q.

Let,

Let the zeroes of polynomial be   and  then,

Sum of zeroes

According to the question, zeroes of   are  and

Sum of zeroes  =

Product of zeroes  =

and q = -6

## If the sum and product of the zeroes of the polynomial  is equal to 10 each, find the value  of 'a' and  'c'

Given, Polynomial

Let the zeroes  of   are  and  , then according to the question

Sum of zeroes,  ()  = Product of zeroes,  = 10

Now,

and

Hence

## If     and  are zeroes of the polynomial  then form a quadratic polynomial whose zeroes are    and

Given, p (x) = 3x2 -4x-7 and  and   are its zeroes.

For the new polynomial

Required quadratic polynomial = x2 - ( Sum of zeroes ) x + Product of zeroes

## Show that    and    are the zeroes of the polynomial  and verify the  relationship between  zeroes and  coefficients of the polynomial.

=1+2-3 =0

and

= 9 - 6 -3 = 0

are zeroes of polynomial

Relation between zeroes and coeff. of polynomial is verified.

## Polynomial    is exactly divisible by  , then find the value of p and q.

Factors  of

......................................(1)

Let

If p(x) is exactly divisible by , then x = -4  and  x = -3 are zeroes of  p(x) [ from eq (1)]

But  p (-4) = 0

....................................(2)

and

but p(-3) =0.

........................................(3)

On solving  eq (2) and  eq (3) by elimination method, we get p =-35

On Putting the value of  p  in eq  (1)

Hence, p = -35, q = -60

## If   and    are the zeros of the polynomial    find the polynomial whose zeroes  are      and

Since  and  are the zeroes of the cubic polynomial

then

and

But required polynomial = x2 - (Sum of the zeroes)x + product of the zeroes

## If   and  are the  zeroes of the polynomial   , find the value of  :

Given   are the zeroes  of

and

Now,

## If the squared difference of the zeroes of the quadratic polynomial  is equal to 144, find the value of p.

The given quadratic polynomial is   Let  and  be the zeroes of the given quadratic polynomial

and

Given

Thus the value of p is

## A two digit number is four times the sum of the digits. It is also equal to 3 times the product of digit. Find the number.

Let units digits and tens digits of the two digit number be x and y

Number  is  10 y + x

According to question,

Also,

or y=2

Rejecting y = 0 as the number can not be zero.

Required number is  24.

## In given figure, the graph of a polynomial p(x) is shown. Calculate the numbers of zeroes of p(x). The numbers of zeroes of p(x) is 1.

## Calculate the zeroes of the polynomial

Let,

The zeroes are

Hence, zeroes of the polynomial are

## If sum of the zeroes of the quadratic polynomial   is  3, then find the value of k.

## If -1 is a zero of the polynomial    then calculate the other zero.

Let the other zero be k, then

## Find all the zeroes of

i,e.,

Hence, zeroes are 0 & 2

## Find  a quadratic polynomial, the sum  and product of whose zeroes are 6 and 9 respectively  Hence find the zeroes.

Sum of zeroes  = 6, product of zeroes = 9

Also,

Hence zeroes are 3,3

## Form a quadratic polynomial p(x) with 3 and 2/5  as sum and product of its zeroes, respectively .

According to the question,

sum of zeroes = 3

Product of zeroes =

(Sum of zeroes) +product of zeroes

## If p,q are zeroes of polynomial  find the value of

We know that,

## Find the condition that zeroes of polynomial  are reciprocal of each other.

So, required condition is, c = a

## Find the values of a and b , if they are the zeroes of polynomial

Product of zeroes =

then

## A policeman and a thief are equidistant from the jewel box. Upon considering the jewel box as origin, the position of a policeman is (0,5). If the ordinate of the position of thief is zero, then what will be the position of the thief?

The position of thief = (5,0) or (-5,0) ## Create the font for the numbers 0,1,2,.......,9 and those fonts having the same number of angles. Example : The number zero has zero angles. That is no angle for the number zero. ## A point whose y- coordinate is zero and x-coordinate is 5 will lie on  a. y-axis        b.x-axis           c. origin       d. None f theses (5,0) lies on x-axis

31

## Calculate the coordinates of the point at which the circle of centre (2,4) and radius 4 units cuts the y axis.

Centre of the circle =(2,4). radius =4

Equation of the circle =

The coordiantes of the point where the circle cuts the y axis is zero.

Coordinates of the point where the circle cuts the y axis =

## In the figure, equation of the line joining the points A and B is x+2y=10. P and Q are points on this line. a. Find the coordinates of the points A and B. b. Find the coordinates of the points P that divides the line AB in the ratio 2:3. c. If AQ: BQ =2:3, find the coordinates of the point Q.

a. Equation of the line AB is x+2y=10

Since the y coordinates of the point A is zero,

Coordinates of A = (10,0)

Since the x coordinates of the point B is sero, 0+2y=10,y=5

Coordinates of B =(0,5)

b. AP : PB =2:3

x coordinates of P =

y coordinates of p =

Coordinates of P=(6,2)

c. AQ: BQ=2:3 Then BA: AQ=1:2

if x is the x coordinates of Q.

If y is the y coordinates of Q.

15+y-5=0,y=5-15=-10

Coordinates of Q =(30,-10)

## Check whether x-3 is a factor of the polynomial .

,

since p(3) is not equal to zero, x-3 is not a factor of p(x).