Q. 8

# Give examples of polynomials p(x), g(x), q(x) and r(x) satisfying the Division Algorithm

p(x) = g(x).q(x) + r(x),deg r(x)<deg g(x)

And also satisfying

(i) deg p(x) = deg q(x) + 1

(ii)deg q(x) = 1

(iii) deg q(x) = deg r(x) + 1

Answer :

(i)Let p(x) = 12x^{2} + 8x + 25, g(x) = 4,

q(x) = 3x^{2} + 2x + 6 , r(x) = 0

Here, degree p(x) = degree q(x) = 2

Now, g(x).q(x) + r(x) = (3x^{2} + 2x + 6)×4 + 1

**=** 12x^{2} + 8x + 24 + 1

= 12x^{2} + 8x + 25

(ii) Let p(x) = t^{3} + t^{2} – 2t, g(x) = t^{2} + 2t,

q(x) = t – 1 , r(x) = 0

Here, degree q(x) = 1

Now, g(x).q(x) + r(x) = (t^{2} + 2t)×(t – 1) + 0

**=** t^{3} – t^{2} + 2t^{2} – 2t

= t^{3} + t^{2} – 2t

(iii) Let p(x) = x^{3} + x^{2} + x + 1 , g(x) = x^{2} – 1,

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

Here, degree q(x) = degree r(x) + 1 = 1

Now, g(x).q(x) + r(x) = (x^{2} – 1)×(x + 1) + 2x + 2

**=** x^{3} + x^{2} – x – 1 + 2x + 2

= x^{3} + x^{2} + x + 1

Rate this question :

Divide the polynomial p(x) by the polynomial g(x) and find the quotient q(x) and remainder r(x) in each case :

p(x) = x^{3} – 3x^{2} + 4x + 2 , g(x) = x – 1

Divide the polynomial p(x) by the polynomial g(x) and find the quotient q(x) and remainder r(x) in each case :

p(x) = x^{3} – 3x^{2}– x + 3, g(x) = x^{2} – 4x + 3

When a polynomial p(x) is divided by (2x + 1), is it possible to have (x - 1) as a remainder? Justify your answer.

RS Aggarwal - MathematicsDivide the polynomial p(x) by the polynomial g(x) and find the quotient q(x) and remainder r(x) in each case :

p(x) = x^{4} + 2x^{3} – 3x^{2} + x – 1, g(x) = x – 2

Find all the zeroes of the polynomial given below having given numbers as its zeroes.

x^{4} – 6x^{3} – 26x^{2} + 138x – 35;2±√3

p(x) = x^{6} + 3x^{2} + 10 and g(x) = x^{3} + 1

p(x) = x^{4} + 1, g(x) = x + 1

Verify that are the zeroes of the cubic polynomial p(x) = 3x^{2} – 5x^{2} – 11x – 3 and then verify the relationship between the zeroes and the coefficients.

Find all the zeros of the polynomial, if two of its zeros are √3 and .

RD Sharma - Mathematics