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in Polynomials by (75 points)
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Apply the division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) as given below:

(i) p(x) 2x2 – 13x + 13; g(x) 3 + x 

(ii) p(x) x3- x2 + 4x – 8; g(x)  x + 3

(iii) p(x) 2x3 + 6x2 – x – 1; g(x)  x2 – 3 

(iv) p(x) 3x3 + 8x2 + 4; g(x)  7 – x2

(v) p(x)  x3 + 12x2 + 7x; g(x)  3 – 2x – x2

by (75 points)
Guys, pls help me solve it. It's been 3 hours, and it's urgent
by (75 points)
Please answer anyone, it's urgent

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1 Answer

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(i) p(x) = 2x2 - 13x + 13, g(x) = 3 + x

∵ 2x2 - 13x + 13 = (2x - 19) (3 + x) + 70

Remainder theorem is 

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

∴ q(x) = 2x - 19

r(x) = 70

(ii) p(x) = x3 - x2 + 4x - 8, g(x) = x + 3

∵ x3 - x2 + 4x - 8 = (x2 - 4x + 16) (x + 3) - 56

∴ q(x) = x2 - 4x + 16

r(x) = -56

(iii) p(x) = 2x3 + 6x2 - x - 1, g(x) = x2 - 3

∵ 2x3 + 6x2 - x  - 1 = (2x + 6) (x2 - 3) + 5x + 17

∴ q(x) = 2x + 6

r(x) = 5x + 17

(iv) p(x) = 3x3 + 8x2 + 4, g(x) = 7 - x2

∵ 3x3 + 8x2 + 4 = (- 3x - 8) (7 - x2) + 21x + 6

\(\)∴ q(x) = -3x - 8 = - (3x + 8)

r(x) = 21x + 6

(v) p(x) = x3 + 12x2 + 7x, g(x) = 3 - 2x - x2

∵ x3 + 12x2 + 7x = (-x - 10) (3 - 2x - x2) -10x + 30

 \(\)∴ q(x) = -x - 10 = -(x + 10)

r(x) = -10x + 30

by (75 points)
Thank you

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