Q4. Divide (3 – x + 2x² + x³ – 3x4 ) by (2 – x ) and verify by division algorithm
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Let p(x)=(x³+2x²-x-12+3)=(x³+2x²-x-9)
According to remainder theorem when p(x) is divided by (2-x), then the remainder will be p(2)
p(x)=(x³+2x²-x-9)
p(2)=(2)³+2(2)²-2-9
8+8-11
16-11=5
Therefore, remainder will be 5.
Verification by division algorithm
Dividend=divisor*quotient+remainder
(x³+2x²-x-9)=(2-x)(-x²-4x-7)+5
(x³+2x²-x-9)=2(-x²-4x-7)-x(-x²-4x-7)+5
(x³+2x²-x-9)=-2x²-8x-14+x³+4x²+7x+5
(x³+2x²-x-9)=x³+2x²-x-9
Thus, verified
According to remainder theorem when p(x) is divided by (2-x), then the remainder will be p(2)
p(x)=(x³+2x²-x-9)
p(2)=(2)³+2(2)²-2-9
8+8-11
16-11=5
Therefore, remainder will be 5.
Verification by division algorithm
Dividend=divisor*quotient+remainder
(x³+2x²-x-9)=(2-x)(-x²-4x-7)+5
(x³+2x²-x-9)=2(-x²-4x-7)-x(-x²-4x-7)+5
(x³+2x²-x-9)=-2x²-8x-14+x³+4x²+7x+5
(x³+2x²-x-9)=x³+2x²-x-9
Thus, verified
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