Q. 8

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

(i)Let p(x) = 12x2 + 8x + 25, g(x) = 4,

q(x) = 3x2 + 2x + 6 , r(x) = 0

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

Now, g(x).q(x) + r(x) = (3x2 + 2x + 6)×4 + 1

= 12x2 + 8x + 24 + 1

= 12x2 + 8x + 25

(ii) Let p(x) = t3 + t2 – 2t, g(x) = t2 + 2t,

q(x) = t – 1 , r(x) = 0

Here, degree q(x) = 1

Now, g(x).q(x) + r(x) = (t2 + 2t)×(t – 1) + 0

= t3 – t2 + 2t2 – 2t

= t3 + t2 – 2t

(iii) Let p(x) = x3 + x2 + x + 1 , g(x) = x2 – 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) = (x2 – 1)×(x + 1) + 2x + 2

= x3 + x2 – x – 1 + 2x + 2

= x3 + x2 + x + 1

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