Worksheet: Real and Complex Roots of Polynomials

In this worksheet, we will practice finding the number and type of roots of polynomials and finding unknown coefficients if the roots are given.


If π‘Ž+𝑏𝑖 is a root of the polynomial 𝑓(π‘₯), what is the value of 𝑓(π‘Ž+𝑏𝑖)?


Does the polynomial π‘Žπ‘§+𝑏𝑧+𝑐𝑧+𝑑𝑧+𝑒𝑧+π‘“οŠ«οŠͺ, where π‘Ž is nonzero and all the coefficients are real, have at least one real root?

  • Ano
  • Byes
  • Ccannot say


If π‘Ž+𝑏𝑖 is a root of the equation 𝑓(π‘₯)=0, where 𝑓(π‘₯) is a polynomial with real coefficients, which other complex number must also be a root?

  • Aπ‘βˆ’π‘Žπ‘–
  • B𝑏+π‘Žπ‘–
  • Cβˆ’π‘Žβˆ’π‘π‘–
  • Dβˆ’π‘Ž+𝑏𝑖
  • Eπ‘Žβˆ’π‘π‘–


Is it possible for a polynomial with real coefficients to have exactly 3 non-real roots?

  • Ano
  • Byes


How many roots does the polynomial (3π‘₯βˆ’1)(π‘₯+4π‘₯βˆ’2) have?


How many real roots could the polynomial 𝑝(π‘₯)=π‘Žπ‘₯+𝑏π‘₯+𝑐π‘₯+𝑑π‘₯+𝑒π‘₯+π‘“οŠ«οŠͺ have given that π‘Ž,𝑏,𝑐,𝑑,𝑒, and 𝑓 are all real?

  • Aonly 2
  • B5, 3, or 1
  • Conly 1
  • D4 or 2
  • E4, 2, or 1


Determine the type of the roots of the equation (π‘₯βˆ’10)(π‘₯+10)=2(π‘₯+8)(π‘₯+6).

  • Acomplex and not real
  • Breal and equal
  • Creal and different


Determine the type of the roots of the equation π‘₯+4π‘₯+1=3.

  • Areal and different
  • Bcomplex and not real
  • Creal and equal


Solve the equation π‘₯+1=0, π‘₯βˆˆβ„‚.

  • Aο―βˆ’12+√32𝑖,βˆ’12βˆ’βˆš32𝑖,1
  • B12+√32𝑖,βˆ’12+√32𝑖,βˆ’1
  • C12+√32𝑖,βˆ’1,12βˆ’βˆš32𝑖
  • D12+√32𝑖


Given that βˆ’2 is one of the roots of the equation π‘₯+6π‘₯+20=0, find the other two roots.

  • A2Β±6𝑖
  • B6Β±2𝑖
  • C1Β±3𝑖
  • D3±𝑖


Find the solution set of βˆ’π‘₯+16=0οŠͺ in the set of complex numbers.

  • A{2,βˆ’2,4𝑖,βˆ’4𝑖}
  • B{2,βˆ’2}
  • C{4,βˆ’4𝑖}
  • D{2,2𝑖}
  • E{2,βˆ’2,2𝑖,βˆ’2𝑖}


Given that 𝑖 is one of the roots of the equation π‘₯βˆ’5π‘₯+π‘₯βˆ’5=0, find the other two roots.

  • Aβˆ’1 and 5
  • Bβˆ’π‘– and 0
  • Cβˆ’π‘– and βˆ’5
  • Dβˆ’1 and βˆ’5
  • Eβˆ’π‘– and 5


Given that 2+π‘–βˆš3 is a root of π‘₯βˆ’12π‘₯+55π‘₯βˆ’120π‘₯+112=0οŠͺ, find all the roots.

  • Aπ‘₯=4, 2+π‘–βˆš3, 2βˆ’π‘–βˆš3
  • Bπ‘₯=βˆ’4, 3+π‘–βˆš2, 3βˆ’π‘–βˆš2
  • Cπ‘₯=βˆ’4, 2+π‘–βˆš3, 2βˆ’π‘–βˆš3
  • Dπ‘₯=4, 3+π‘–βˆš2, 3βˆ’π‘–βˆš2
  • Eπ‘₯=4, 2+√11, βˆ’2βˆ’βˆš11

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