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  • Answer the following and justify: Can x^2 – 1 be the quotient on division of x^6 + 2x^3 + x – 1 by a polynomial in x of degree 5?

(i)Answer the following and justify:

Can x^2 - 1 be the quotient on division of x^6 + 2x^3 + x - 1 by a polynomial in x of degree 5?

(ii)Answer the following and justify:

What will the quotient and remainder be on division of ax^2 + bx + c \ by \ px^3 + qx^2 + rx+ s, p\neq 0?ax^2 + bx + c \ by \ px^3 + qx^2 + rx+ s, p\neq 0?

(iii)Answer the following and justify:

If on division of a polynomial p (x) by a polynomial g (x), the quotients zero, what is the relation between the degrees of p (x) and g (x)?

(iv) Answer the following and justify:

If on division of a non-zero polynomial p (x) by a polynomial g (x), the remainder is zero, what is the relation between the degrees of p (x) and g (x)?

(v) Answer the following and justify:

Can the quadratic polynomial  x^2 + kx + khave equal zeroes for some odd integer k > 1?

 

 

Answers (1)

(i) Answer. [false]

Solution.        

Let divisor of a polynomial in x of degree 5 is = x^5 + x + 1

Quotient = x^2 - 1

Dividend = x^6 + 2x^3 + x - 1

According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) \neq 0, then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by

p(x) = g(x) \times q(x) + r(x)

i,e. Dividend = Divisor × Quotient + Remainder

= (x^5 + x + 1)(x^2 - 1) + Remainder

= x^7 - x^5 + x^3 + x + x^2 - 1 + Remainder

Here it is of degree seven but given dividend is of degree six.

Therefore x^2 - 1can not be the quotient of x^6 + 2x^3 + x - 1 because division algorithm is not satisfied.

Hence, given statement is false.

(ii) Here dividend is ax^2 + bx + c

and divisor is px^3 + qx^2 + rx + s

According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) ¹ 0 then the relation among p(x), g(x) quotient q(x) and remainder r(x) is given by

p(x) = g(x) \times q(x) + r(x)

where degree of r(x) < degree of g(x).

i.e. Dividend = Devisor × Quotient + Remainder

Here degree of divisor is greater than degree of dividend therefore.

According to division algorithm theorem ax^2 + bx + c is the remainder and quotient will be zero.

That is remainder = ax^2 + bx + c

Quotient = 0

(iii) Division algorithm theorem :- According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) is then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by

p(x) = g(x) × q(x) + r(x)

where degree of r(x) < degree of g(x)

i.e. Dividend = Division × Quotient + Remainder

In the given statement it is given that on division of a polynomial p(x) by a polynomial g(x), the quotient is zero.

The given condition is possible only when degree of divisor is greater than degree of dividend

i.e. degree of g(x) > degree of p(x).

(iv)

Division algorithm theorem :- According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) is then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by

p(x) = g(x) × q(x) + r(x)          ….(1)

where degree of r(x) < degree of g(x)

According to given statement on division of a non-zero polynomial p(x) by a polynomial g(x) then remainder r(x) is zero then equation (1) becomes

p(x) = g(x) × q(x)                    ….(2)

From equation (2) we can say g(x) is a factor of p(x) and degree of g(x) may be less than or equal to p(x).

 

(v)

Answer.  [false]

 Polynomial : It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s)

Quadratic polynomial : when the degree of polynomial is two then the polynomial is called quadratic polynomial.

Let p(x)=x^2 + kx + k

It is given that zeroes of p(x) has equal and we know that when any polynomial having equal zeroes than their discriminate is equal to zero

i.e. d = b^2 - 4ac = 0

p(x)=x^2 + kx + k

here, a = 1, b = k, c = k

\therefore d = (k)^2 - 4 \times 1 \times k = 0
k^2 - 4k = 0

k(k -4) = 0

k = 0, k = 4

Hence, the given quadratic polynomial x^2 + kx + k have equal zeroes only when the values of k will be 0 and 4.

Hence, given statement is not correct.

 

Posted by

infoexpert21

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