Worksheet: Descartes’ Rule of Signs

In this worksheet, we will practice using Descartes’ 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‘₯)=3π‘₯+2π‘₯βˆ’3π‘₯+π‘₯+6π‘₯+2οŠͺ.

  • Aβ„Ž(π‘₯) has either 3 or 1 positive zeros and 0 negative zeros.
  • Bβ„Ž(π‘₯) has either 2 or 0 positive zeros and 1 negative zero.
  • Cβ„Ž(π‘₯) has either 2 or 0 positive zeros and either 3 or 1 negative zeros.
  • Dβ„Ž(π‘₯) has either 3 or 1 positive zeros and either 2 or 0 negative zeros.
  • Eβ„Ž(π‘₯) has 0 positive zeros and either 3 or 1 negative zeros.

Q2:

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

  • A𝑓(π‘₯) has 0 positive zeros and either 3 or 1 negative zeros.
  • B𝑓(π‘₯) has 1 positive zero and 0 negative zeros.
  • C𝑓(π‘₯) has either 3 or 1 positive zeros and 0 negative zeros.
  • D𝑓(π‘₯) has either 2 or 0 positive zeros and one negative zero.
  • E𝑔(π‘₯) has 1 positive zero and either 2 or 0 negative zeros.

Q3:

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

  • A𝑔(π‘₯) has either 3 or 1 positive zeros and 1 negative zero.
  • B𝑔(π‘₯) has either 2 or 0 positive zeros and either 2 or 0 negative zeros.
  • C𝑔(π‘₯) has 0 positive zeros and either 2 or 0 negative zeros.
  • D𝑔(π‘₯) has 1 positive zero and either 3 or 1 negative zeros.
  • E𝑔(π‘₯) has either 2 or 0 positive zeros and 0 negative zeros.

Q4:

How many roots does the polynomial 5π‘₯+4π‘₯βˆ’1 have?

Q5:

Which of the following lists contain all the roots of 4π‘₯βˆ’10π‘₯βˆ’76π‘₯+70π‘₯+300οŠͺ?

  • Aβˆ’3,βˆ’2,5,βˆ’52
  • B3,2,βˆ’5,βˆ’52
  • Cβˆ’3,βˆ’2,5,52
  • Dβˆ’3,2,5,52

Q6:

Find, by factoring, the zeros of the function 𝑓(π‘₯)=12π‘₯βˆ’14π‘₯βˆ’6.

  • Aβˆ’13,βˆ’32
  • Bβˆ’23,34
  • C13,βˆ’32
  • D23,34
  • Eβˆ’13,32

Q7:

Is (π‘₯βˆ’π‘Ž) a factor of 𝑓(π‘₯) given that 𝑓(π‘Ž)=3?

  • Ayes
  • Bno

Q8:

Find, by factoring, the zeros of the function 𝑓(π‘₯)=6π‘₯+33π‘₯βˆ’63π‘₯οŠͺ.

  • Aβˆ’7,0,32
  • B7,0,βˆ’32
  • C3,0,βˆ’72
  • Dβˆ’7,0,βˆ’32
  • Eβˆ’3,0,72

Q9:

Which of the following are roots of 6π‘₯+43π‘₯βˆ’169π‘₯βˆ’988π‘₯+1,948π‘₯+1,680οŠͺ?

  • A4, 25 and βˆ’23
  • B6, 52 and 23
  • Cβˆ’7, βˆ’52 and βˆ’23
  • Dβˆ’6, 52 and βˆ’23
  • E4, βˆ’6 and 7

Q10:

Is (π‘₯βˆ’π‘Ž) a factor of 𝑓(π‘₯) given that 𝑓(π‘Ž)=0?

  • Ayes
  • Bno

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