(1) 2x2-5/6x=1/12
(2) using remainder theorem;find the remainder when x3+3x2+3x+1 is divided by (x-1/2
(3) using factor theorem , show that (x-3) is a factor of p(x); x3+x2-17x+15
(4) x8-y8 with formula

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Solution

(2) Using remainder theorem;find the remainder when x3+3x2+3x+1 is divided by (x-1/2)
ans.
$x - \frac{1}{2} = 0 x = \frac{1}{2} P u t t h i s i n t h e g i v e n p o l y n o m i a l p (x) = x^{3} + 3 x^{2} + 3 x +^{3} 1 R e m a i n d e r = p (\frac{1}{2}) = {(\frac{1}{2})}^{3} + 3 {(\frac{1}{2})}^{2} + 3 (\frac{1}{2}) + 1 = \frac{1}{8} + \frac{3}{4} + \frac{3}{2} + 1 = \frac{1 + 6 + 12 + 8}{8} = \frac{27}{8}$

(3) using factor theorem , show that (x-3) is a factor of p(x) = x³+x²-17x+15

x-3=0
x=3
By factor theorem if (x - 3) is a factor of p(x), then p(3) = 0
now, p(3) = 3³+3²-17(3)+15=27+9-51+15 = 0

Hence (x-3) is a factor.

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