Answer:
Remainder when p(x) is divided by (x+2) is -29
Step-by-step explanation:
p(x) = [tex]x^{3} - 2x^{2} + 8x + k[/tex]
When p(x) is divided by (x-2), remainder is 19.
p(x - 2 = 0) gives the remainder when p(x) is divided by (x-2)
x - 2 = 0
x = 2
p(x-2=0) = p(2) = [tex]2^{3} - 2(2^{2}) + 8(2) + k[/tex] = 19
8 - 8 + 16 + k = 19
k = 3
p(x) = [tex]x^{3} - 2x^{2} + 8x + 3[/tex]
p(x + 2 = 0) gives the remainder when p(x) is divided by (x+2)
x + 2 = 0
x = -2
p(x+2=0) = p(-2) = [tex](-2)^{3} - 2((-2)^{2}) + 8(-2) + 3[/tex]
p(-2) = - 8 - 8 - 16 + 3 = -29
Remainder when p(x) is divided by (x+2) is -29
✩ Given Polynomial : x³ - 2x² + 8x + k
when p(x) is divided by (x - 2) it gives 19 as remainder. HENCE WE CAN SAY.
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[tex]:\implies\sf p(x)=x^3-2x^2+8x+k\\\\\\:\implies\sf p(x-2=0)=x^3-2x^2+8x+k\\\\\\:\implies\sf p(x=2)=x^3-2x^2+8x+k\\\\\\:\implies\sf p(2)=(2)^3-2(2)^2+8(2)+k\\\\\\:\implies\sf 19 = 8 - 8 + 16 + k\\\\\\:\implies\sf 19 - 16 = k\\\\\\:\implies\sf k = 3[/tex]
[tex]\rule{150}{1}[/tex]
☯ [tex]\underline{\boldsymbol{According\: to \:the\: Question\:now :}}[/tex]
[tex]:\implies\sf p(x)=x^3-2x^2+8x+k\\\\\\:\implies\sf p(x+2=0)=x^3-2x^2+8x+k\\\\\\:\implies\sf p(x=-\:2)=x^3-2x^2+8x+k\\\\\\:\implies\sf p(-\:2)=(-\:2)^3-2(-\:2)^2+8(-\:2)+3\\\\\\:\implies\sf p(-\:2) = - \:8 - \:8 - \:16 + 3\\\\\\:\implies\underline{\boxed{\sf p(-\:2) = - \:29}}[/tex]
⠀
[tex]\therefore\:\underline{\textsf{Hence, required remainder will be \textbf{- 29}}}.[/tex]
