divide P(x) by g(x)p(x)=x^4-3x^2+4x+5g(x)=x^2+1-x

1.	divide P(x) by g(x)p(x)=x^4-3x^2+4x+5g(x)=x^2+1-x
Answer» {tex}p ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 4 x + 5 , \\quad g ( x ) = x ^ { 2 } + 1 - x{/tex}p(x) is in standard form.g(x), in standard form, is x2 - x + 1.Now, we apply the division algorithm to the given polynomial p(x) and g(x).We stop here since degree (8) = 0 < degree (x2 - x + 1)So, quotient = x2 + x - 3, remainder = 8Therefore,{tex}\\text { Quotient } \\times \\text { Divisor } + \\text { Remainder }{/tex}{tex}= \\left( x ^ { 2 } + x - 3 \\right) \\left( x ^ { 2 } - x + 1 \\right) + 8{/tex}{tex}= x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + x ^ { 3 } - x ^ { 2 } + x - 3 x ^ { 2 } + 3 x - 3 + 8{/tex}{tex}= x ^ { 4 } - 3 x ^ { 2 } + 4 x + 5{/tex}= DividendTherefore, the division algorithm is verified.

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