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**EXERCISE 2.3****Ques 1: Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :(i) p(x) = x ^{3} - 3x^{2} + 5x - 3, g(x) = x^{2} - 2**

divisor g(x) = x

âˆ´ We have

Thus, the quotient = (x - 3) and remainder = (7x - 9)

(ii) Here, dividend p(x) = x^{4 }- 3x^{2} + 4x + 5

and divisor g(x) = x^{2} + 1 - x

= x^{2} - x + 1

âˆ´ We have

Thus, the quotient is (x^{2} + x - 3) and remainder = 8

(iii) Here, dividend, p(x) = x^{4} - 5x + 6

and divisor, g(x) = 2 - x^{2}

= - x^{2} + 2

âˆ´ We have

Thus, the quotient = â€“x^{2} â€“ 2 and remainder = â€“5x + 10.**Ques 2: Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:****Sol. **(i) Dividing 2t^{4} + 3t^{3} - 2t^{2} - 9t - 12 by t^{2} - 3, we have:

âˆµ Remainder = 0

âˆ´ (t^{2} - 3) is a factor of 2t^{4} + 3t^{3} - 2t^{2} - 9t - 12.

(ii) Dividing 3x^{4} + 5x^{3} - 7x^{2} + 2x + 2 by x^{2} + 3x + 1, we have:

âˆµ Remainder = 0.

âˆ´ x^{2} + 3x + 1 is a factor of 3x^{4} + 5x^{3} - 7x^{2 }+ 2x + 2.

(iii) Dividing x^{5} - 4x^{3} + x^{2} + 3x + 1 by x^{3} - 3x + 1, we get:

âˆ´ The remainder = 2, i.e., remainder â‰ 0

âˆ´ x^{3} - 3x + 1 **is not a factor** of x^{5 }- 4x^{3} + x^{2} + 3x + 1.**Ques 3: Obtain all other zeroes of 3x ^{4} + 6x^{3} - 2x^{2} - 10x - 5, if two of its zeroes **

Î± = and Î² = -

âˆ´ is a factor of p (x).

Now, let us divide 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5 by .

For p (x) = 0, we have

i.e.,

or 3x + 3 = 0 â‡’ x = - 1

or x + 1 = 0 â‡’ x = - 1

Thus, all the other zeroes of the given polynomial are - 1 and - 1.**Ques 4: On dividing x ^{3 }- 3x^{2} + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and -2x + 4 respectively. Find g(x).**

Dividend p(x) = x

Divisor = g(x)

Quotient = (x - 2)

Remainder = (-2x + 4)

Since,

(Quotient Ã— Divisor) + Remainder = Dividend

âˆ´ [(x âˆ’ 2) Ã— g(x)] + [(âˆ’2x + 4)] = x^{3} âˆ’ 3x^{2} + x + 2

â‡’ (x âˆ’ 2) Ã— g(x)= x^{3} âˆ’ 3x^{2} + x + 2 âˆ’ (âˆ’2x + 4)

= x^{3} âˆ’ 3x^{2} + x + 2 + 2x âˆ’4

= x^{3} - 3x^{2} + 3x - 2

âˆ´

Now, dividing x^{3} - 3x^{2 }+ 3x - 2 by x - 2, we have

âˆ´

Thus, the required divisor g(x) = x^{2 }- x + 1.**Ques 5: Give example of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and(i) deg p(x) = deg q(x)(ii) deg q(x) = deg r(x)(iii) deg r(x) = 0**

(i) p(x), g(x), q(x), r(x)

deg p(x) = deg q(x)

âˆ´ both g(x) and r(x) are constant terms.

p(x) = 2x

g(x) = 2

q(x) = x

r(x) = 0

(ii) deg q(x) = deg r(x)

âˆ´ this is possible when deg of both q(x) and r(x) should be less than p(x) and g(x).

p(x) = x^{3} + x^{2} + x + 1

g(x) = x^{2} - 1

q(x) = x + 1, r(x) = 2x + 2

(iii) deg r(x) is 0.

This is possible when product of q(x) and g(x) form a polynomial whose degree is equal to degree of p(x) and constant term.

NOTE:We have given one example for each of the above cases, however, there can be several examples for them.

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