Worksheet: Descartes's Rule of Signs

In this worksheet, we will practice using Descartes’s rule of signs to get all possible numbers of positive and negative real roots.

Q1:

Using Descartes’s rule of signs, find which of the following describes the possible real zeros of β„Ž ( π‘₯ ) = 3 π‘₯ + 2 π‘₯ βˆ’ 3 π‘₯ + π‘₯ + 6 π‘₯ + 2  οŠͺ   .

  • A β„Ž ( π‘₯ ) has 0 positive zeros and either 3 or 1 negative zeros.
  • B β„Ž ( π‘₯ ) has either 3 or 1 positive zeros and 0 negative zeros.
  • C β„Ž ( π‘₯ ) has either 2 or 0 positive zeros and 1 negative zero.
  • D β„Ž ( π‘₯ ) has either 2 or 0 positive zeros and either 3 or 1 negative zeros.
  • E β„Ž ( π‘₯ ) has either 3 or 1 positive zeros and either 2 or 0 negative zeros.

Q2:

Using Descartes’s rule of signs, find which of the following describes the possible real zeros of 𝑓 ( π‘₯ ) = βˆ’ 6 π‘₯ + 3 π‘₯ + 7 π‘₯ βˆ’ 1   .

  • A 𝑓 ( π‘₯ ) has 1 positive zero and 0 negative zeros.
  • B 𝑓 ( π‘₯ ) has 0 positive zeros and either 3 or 1 negative zeros.
  • C 𝑔 ( π‘₯ ) has 1 positive zero and either 2 or 0 negative zeros.
  • D 𝑓 ( π‘₯ ) has either 3 or 1 positive zeros and 0 negative zeros.
  • E 𝑓 ( π‘₯ ) has either 2 or 0 positive zeros and one negative zero.

Q3:

Using Descartes’s rule of signs, find which of the following describes the possible real zeros of 𝑔 ( π‘₯ ) = βˆ’ π‘₯ βˆ’ 3 π‘₯ + 4 π‘₯ + 9 π‘₯ βˆ’ 4 οŠͺ   .

  • A 𝑔 ( π‘₯ ) has either 3 or 1 positive zeros and 1 negative zero.
  • B 𝑔 ( π‘₯ ) has 1 positive zero and either 3 or 1 negative zeros.
  • C 𝑔 ( π‘₯ ) has either 2 or 0 positive zeros and either 2 or 0 negative zeros.
  • D 𝑔 ( π‘₯ ) has 0 positive zeros and either 2 or 0 negative zeros.
  • E 𝑔 ( π‘₯ ) has either 2 or 0 positive zeros and 0 negative zeros.

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