Worksheet: Writing a Polynomial Function given Zeros

In this worksheet, we will practice using zeros to write a polynomial function of the least degree with integral coefficients.

Q1:

Find, in its simplest form, the quadratic equation whose roots are 3 and 8 .

  • A 𝑥 1 1 𝑥 + 2 4 = 0 2
  • B 𝑥 + 2 4 𝑥 + 1 1 = 0 2
  • C 𝑥 + 1 1 𝑥 2 4 = 0 2
  • D 𝑥 + 1 1 𝑥 + 2 4 = 0 2
  • E 𝑥 2 4 𝑥 1 1 = 0 2

Q2:

Find, in its simplest form, the quadratic equation whose roots are 1 4 𝑖 and 1 4 𝑖 .

  • A 𝑥 + 2 8 = 0 2
  • B 𝑥 1 9 6 = 0 2
  • C 𝑥 2 8 = 0 2
  • D 𝑥 + 1 9 6 = 0 2

Q3:

Find, in its simplest form, the quadratic equation whose roots are 4 + 5 𝑖 and 4 5 𝑖 .

  • A 𝑥 8 𝑥 4 1 = 0 2
  • B 𝑥 8 𝑥 + 4 1 = 0 2
  • C 𝑥 + 8 𝑥 9 = 0 2
  • D 𝑥 + 8 𝑥 + 4 1 = 0 2
  • E 𝑥 8 𝑥 9 = 0 2

Q4:

Which quadratic equation has roots 𝑥 = 2 ± 𝑖 ?

  • A 𝑥 4 𝑥 + 3 = 0 2
  • B 𝑥 + 4 𝑥 + 5 = 0 2
  • C 𝑥 + 4 𝑥 + 3 = 0 2
  • D 𝑥 4 𝑥 + 5 = 0 2
  • E 𝑥 5 𝑥 + 4 = 0 2

Q5:

Find, in its simplest form, the quadratic equation whose roots are 7 2 and 2 7 .

  • A 5 3 𝑥 1 4 𝑥 + 5 3 = 0 2
  • B 1 4 𝑥 + 5 3 𝑥 + 1 4 = 0 2
  • C 5 3 𝑥 + 1 4 𝑥 + 5 3 = 0 2
  • D 1 4 𝑥 5 3 𝑥 + 1 4 = 0 2
  • E 1 4 𝑥 5 3 𝑥 + 5 3 = 0 2

Q6:

Find, in its simplest form, the quadratic equation whose roots are 5 + 2 and 5 2 .

  • A 𝑥 + 1 0 𝑥 + 2 3 = 0 2
  • B 𝑥 1 0 𝑥 + 2 1 = 0 2
  • C 𝑥 1 0 𝑥 + 2 7 = 0 2
  • D 𝑥 1 0 𝑥 + 2 3 = 0 2
  • E 𝑥 + 1 0 𝑥 + 2 7 = 0 2

Q7:

Find, in its simplest form, the quadratic equation whose roots are 2 5 + 3 and 2 5 3 .

  • A 𝑥 + 4 5 𝑥 + 1 7 = 0 2
  • B 𝑥 4 5 𝑥 + 1 1 = 0 2
  • C 𝑥 4 5 𝑥 1 9 = 0 2
  • D 𝑥 + 4 5 𝑥 + 1 1 = 0 2
  • E 𝑥 + 4 𝑥 + 1 1 = 0 2

Q8:

Find, in its simplest form, the quadratic equation whose roots are 2 + 3 7 𝑖 and 2 3 7 𝑖 .

  • A 𝑥 4 𝑥 5 9 = 0 2
  • B 𝑥 + 4 𝑥 + 6 7 = 0 2
  • C 𝑥 4 𝑥 5 9 𝑖 = 0 2
  • D 𝑥 4 𝑥 + 6 7 = 0 2
  • E 𝑥 + 4 7 𝑥 + 6 7 = 0 2

Q9:

Find, in its simplest form, the quadratic equation whose roots are 2 + 2 𝑖 1 + 𝑖 and 2 + 4 𝑖 2 𝑖 .

  • A 𝑥 4 = 0 2
  • B 𝑥 + 4 𝑖 = 0 2
  • C 𝑥 + 4 𝑥 = 0 2
  • D 𝑥 + 4 = 0 2

Q10:

Find, in its simplest form, the quadratic equation whose roots are 2 and 8.

  • A 𝑥 + 1 0 𝑥 + 1 6 = 0 2
  • B 𝑥 + 1 6 𝑥 1 0 = 0 2
  • C 𝑥 1 0 𝑥 1 6 = 0 2
  • D 𝑥 1 0 𝑥 + 1 6 = 0 2
  • E 𝑥 1 6 𝑥 + 1 0 = 0 2

Q11:

Find, in its simplest form, the quadratic equation whose roots are 9 1 2 𝑖 and 9 + 1 2 𝑖 .

  • A 𝑥 + 1 8 𝑥 2 2 5 = 0 2
  • B 𝑥 + 1 8 𝑥 + 2 2 5 = 0 2
  • C 𝑥 1 8 𝑥 6 3 = 0 2
  • D 𝑥 1 8 𝑥 + 2 2 5 = 0 2
  • E 𝑥 + 1 8 𝑥 6 3 = 0 2

Q12:

Find, in its simplest form, the quadratic equation whose roots are 1 2 𝑖 and 1 2 𝑖 .

  • A 𝑥 + 2 4 = 0 2
  • B 𝑥 1 4 4 = 0 2
  • C 𝑥 2 4 = 0 2
  • D 𝑥 + 1 4 4 = 0 2

Q13:

Find, in its simplest form, the quadratic equation whose roots are 1 + 5 and 1 5 .

  • A 𝑥 + 2 𝑥 4 = 0 2
  • B 𝑥 2 𝑥 2 4 = 0 2
  • C 𝑥 2 𝑥 + 6 = 0 2
  • D 𝑥 2 𝑥 4 = 0 2
  • E 𝑥 + 2 𝑥 + 6 = 0 2

Q14:

Find, in its simplest form, the quadratic equation whose roots are 2 2 + 6 and 2 2 6 .

  • A 𝑥 4 2 𝑥 + 2 = 0 2
  • B 𝑥 + 4 2 𝑥 2 8 = 0 2
  • C 𝑥 + 4 2 𝑥 3 2 = 0 2
  • D 𝑥 4 2 𝑥 2 8 = 0 2
  • E 𝑥 4 𝑥 2 8 = 0 2

Q15:

Find, in its simplest form, the quadratic equation whose roots are 6 7 and 7 6 .

  • A 8 5 𝑥 4 2 𝑥 + 8 5 = 0 2
  • B 4 2 𝑥 + 8 5 𝑥 + 4 2 = 0 2
  • C 8 5 𝑥 + 4 2 𝑥 + 8 5 = 0 2
  • D 4 2 𝑥 8 5 𝑥 + 4 2 = 0 2
  • E 4 2 𝑥 8 5 𝑥 + 8 5 = 0 2

Q16:

Find, in its simplest form, the quadratic equation whose roots are 3 5 and 5 3 .

  • A 3 4 𝑥 1 5 𝑥 3 4 = 0 2
  • B 1 5 𝑥 3 4 𝑥 + 1 5 = 0 2
  • C 3 4 𝑥 + 1 5 𝑥 3 4 = 0 2
  • D 1 5 𝑥 + 3 4 𝑥 + 1 5 = 0 2
  • E 1 5 𝑥 + 3 4 𝑥 3 4 = 0 2

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