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गुणनखण्ड प्रमेय का प्रयोग करके जाँचिये कि g(x), बहुपद f(x) का गुणनखण्ड है या नहीं।

(i) f(x) = x3 – 6x2 -19x + 84 तथा g(x) = x – 7

(ii) f(x) = x3 – 3x2 +4x – 4 तथा g(x) = x – 2

(iii) f(x) = 3x4 + 17x3 + 9x2 – 7x – 10 तथा g(x) = x + 5

(iv) f(x) = 2x3 + 4x + 6 तथा g(x) = x + 1

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(i) f(x) = x3 – 6x2 – 19x + 84 तथा g(x) = x – 7

g(x) = x – 7 = 0 या x = 7 का मान f(x) में रखने पर

f(7) = (7)3 – 6(7)2 – 19(7) + 84

= 343 – 294 – 133 + 84

=427 – 427 = 0

अतः g(x), f(x) का एक गुणनखण्ड है।

(ii) f(x) = x3 – 3x2 + 4x – 4 तथा g(x) = x – 2

g(x) = 0 या x – 2 = 0 या x = 2 रखने पर

f(2) = (2)3 – 3(2)2 + 4(2) – 4

= 8 – 12 + 8 – 4

= 16 – 16 = 0

अतः g(x), f(x) का एक गुणनखण्ड है।

(iii) f(x) = 3x4 + 17x3 + 9x2 – 7x – 10 तथा g(x) = x + 5

g(x) = 0 या x + 5 = 0 या x = -5 रखने पर

f(-5) = 3(-5)4 + 17(-5)3 + 9(-5)2 – 7(-5) – 10

= 3 × 625 – 17 × 125 + 9 × 25 + 35 – 10

= 1875 – 2125 + 225 + 25

= 2125 – 2125 = 0

अतः g(x), f(x) का एक गुणनखण्ड है।

(iv) f(x) = 2x3 + 4x + 6 तथा g(x) = x +1

g(x) = 0 या x + 1 = 0 या x = -1 रखने पर

f(-1) = 2(-1)3 + 4(-1) + 6

= -2 – 4 + 6 = 0

अतः g(x), f (x) का एक गुणनखण्ड है।

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