A polynomial P is given. (a) List all rational possible zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes

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Pulad1971 · Accepted Answer

Step 1 To find the rational possible zeros. Step 2 given that P(x)=3x7−x5+5x4+x3+8 The leading coefficient of P is 3 and the constant term is 8. the possible rational zero of P is =factor of 8factor of 3 The factor of 8 are ±1,±2,±4,±8. The factor of 3 are ±1,±3. Thus the possible rational zeros of P are ±11,±21,±41,±81,±13,±23,±43,±83. Simplifying the fractions ±13,±23,±1,±43,±2,±83,±4,±8 b) Now to determine the possible number of positive and negative real zeros using Descartes rule of signs. Here the polynomial P(x)=3x7−x5+5x4+x3+8 has two variations in signs so it has two or none positive zeros. now P(−x)=−3x7+x5+5x4−x3+8 So P(x) has three sign variations thus P(x) has either three or one negative zeros.

A polynomial P is given. (a) List all rational possible zeros (without

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