Worksheet: Linear Factorization and Conjugate Root Theorems

In this worksheet, we will practice writing a polynomial function given its zeros using linear factorization and conjugate root theorems.

Q1:

Write a polynomial function of the least degree with real coefficients in standard form given that it has 1,2, and 7+2𝑖 as zeros.

  • A𝑓(π‘₯)=π‘₯βˆ’17π‘₯+93π‘₯βˆ’131π‘₯βˆ’106οŠͺ
  • B𝑓(π‘₯)=π‘₯βˆ’17π‘₯βˆ’9π‘₯+131π‘₯βˆ’106οŠͺ
  • C𝑓(π‘₯)=π‘₯+17π‘₯+97π‘₯+187π‘₯+106οŠͺ
  • D𝑓(π‘₯)=π‘₯βˆ’17π‘₯+97π‘₯βˆ’187π‘₯+106οŠͺ
  • E𝑓(π‘₯)=π‘₯+17π‘₯βˆ’13π‘₯βˆ’187π‘₯+106οŠͺ

Q2:

Write a polynomial function of least degree with real coefficients in standard form given that it has βˆ’4 and 2𝑖 (multiplicity 2) as zeros.

  • A𝑓(π‘₯)=π‘₯+4π‘₯+8π‘₯+32π‘₯+16π‘₯+64οŠͺ
  • B𝑓(π‘₯)=π‘₯βˆ’4π‘₯βˆ’8π‘₯+32π‘₯+16π‘₯βˆ’64οŠͺ
  • C𝑓(π‘₯)=π‘₯+4π‘₯βˆ’8π‘₯βˆ’32π‘₯+16π‘₯+64οŠͺ
  • D𝑓(π‘₯)=π‘₯βˆ’4π‘₯+8π‘₯βˆ’32π‘₯+16π‘₯βˆ’64οŠͺ

Q3:

Write a polynomial function of the least degree with real coefficients in standard form given that it has √5+2, βˆ’βˆš5+2 and 3βˆ’π‘– as zeros.

  • A𝑓(π‘₯)=π‘₯βˆ’10π‘₯+35π‘₯βˆ’46π‘₯+10οŠͺ
  • B𝑓(π‘₯)=π‘₯βˆ’10π‘₯+13π‘₯+46π‘₯+10οŠͺ
  • C𝑓(π‘₯)=π‘₯βˆ’10π‘₯+33π‘₯βˆ’34π‘₯βˆ’10οŠͺ
  • D𝑓(π‘₯)=π‘₯βˆ’2π‘₯βˆ’15π‘₯+46π‘₯βˆ’10οŠͺ
  • E𝑓(π‘₯)=π‘₯+10π‘₯+33π‘₯+34π‘₯βˆ’10οŠͺ

Q4:

Consider 𝑔(π‘₯)=π‘₯βˆ’7π‘₯+11π‘₯βˆ’41π‘₯+180οŠͺ.

Write 𝑔(π‘₯) as the product of linear and irreducible quadratic factors.

  • A𝑔(π‘₯)=(π‘₯+4)(π‘₯+5)(π‘₯+π‘₯+4)
  • B𝑔(π‘₯)=(π‘₯+4)(π‘₯+5)(π‘₯+2π‘₯+9)
  • C𝑔(π‘₯)=(π‘₯βˆ’4)(π‘₯βˆ’5)(π‘₯+π‘₯+4)
  • D𝑔(π‘₯)=(π‘₯βˆ’4)(π‘₯βˆ’5)(π‘₯+2π‘₯+9)

Write 𝑔(π‘₯) as the product of linear factors.

  • A𝑔(π‘₯)=(π‘₯+4)(π‘₯+5)ο€»π‘₯+1βˆ’2√2𝑖π‘₯+1+2√2𝑖
  • B𝑔(π‘₯)=(π‘₯+4)(π‘₯+5)ο€Ώπ‘₯+12βˆ’βˆš152𝑖π‘₯+12+√152𝑖
  • C𝑔(π‘₯)=(π‘₯βˆ’4)(π‘₯βˆ’5)ο€»π‘₯+1βˆ’2√2𝑖π‘₯+1+2√2𝑖
  • D𝑔(π‘₯)=(π‘₯βˆ’4)(π‘₯βˆ’5)ο€Ώπ‘₯+12βˆ’βˆš152𝑖π‘₯+12+√152𝑖

List all zeros of 𝑔(π‘₯).

  • Aβˆ’5,βˆ’4,βˆ’12βˆ’βˆš152𝑖,βˆ’12+√152𝑖
  • B4,5,βˆ’12βˆ’βˆš152𝑖,βˆ’12+√152𝑖
  • C4,5,βˆ’1+2√2𝑖,βˆ’1βˆ’2√2𝑖
  • Dβˆ’5,βˆ’4,βˆ’1+2√2𝑖,βˆ’1βˆ’2√2𝑖

Q5:

Consider β„Ž(π‘₯)=5π‘₯βˆ’4π‘₯βˆ’81π‘₯+134π‘₯+30οŠͺ.

Write β„Ž(π‘₯) as the product of linear and irreducible quadratic factors.

  • Aβ„Ž(π‘₯)=(π‘₯βˆ’3)(5π‘₯+1)(π‘₯+2π‘₯βˆ’10)
  • Bβ„Ž(π‘₯)=(π‘₯+3)(5π‘₯βˆ’1)(π‘₯+1+√11)(π‘₯+1βˆ’βˆš11)
  • Cβ„Ž(π‘₯)=(π‘₯βˆ’3)(5π‘₯+1)(π‘₯βˆ’1+√11)(π‘₯βˆ’1βˆ’βˆš11)
  • Dβ„Ž(π‘₯)=(π‘₯βˆ’3)(5π‘₯+1)(π‘₯+1+√11)(π‘₯+1βˆ’βˆš11)
  • Eβ„Ž(π‘₯)=(π‘₯+3)(5π‘₯βˆ’1)(π‘₯+2π‘₯βˆ’10)

List all zeros of β„Ž(π‘₯).

  • A3,βˆ’15
  • Bβˆ’3,15
  • Cβˆ’3,15,βˆ’1βˆ’βˆš11,√11βˆ’1
  • D3,βˆ’15,βˆ’1βˆ’βˆš11,√11βˆ’1
  • E3,βˆ’15,1βˆ’βˆš11,1+√11

Q6:

Consider 𝑓(π‘₯)=π‘₯+3π‘₯βˆ’5π‘₯βˆ’3π‘₯+4οŠͺ.

Write 𝑓(π‘₯) as the product of linear and irreducible quadratic factors.

  • A𝑓(π‘₯)=(π‘₯βˆ’1)(π‘₯βˆ’4)(π‘₯+1)
  • B𝑓(π‘₯)=(π‘₯+4)(π‘₯βˆ’1)
  • C𝑓(π‘₯)=(π‘₯+4)(π‘₯+1)
  • D𝑓(π‘₯)=(π‘₯βˆ’1)(π‘₯+4)(π‘₯+1)
  • E𝑓(π‘₯)=(π‘₯+1)(π‘₯+4)(π‘₯βˆ’1)

List all zeros of 𝑓(π‘₯).

  • Aβˆ’1,βˆ’4,1
  • Bβˆ’4,1
  • Cβˆ’1,4,1
  • Dβˆ’4,βˆ’1

Q7:

Consider π‘˜(π‘₯)=βˆ’3π‘₯βˆ’7π‘₯βˆ’7π‘₯+15π‘₯+50οŠͺ.

Write π‘˜(π‘₯) as the product of linear and irreducible quadratic factors.

  • Aπ‘˜(π‘₯)=βˆ’(π‘₯βˆ’2)(3π‘₯+5)ο€Ήπ‘₯+2π‘₯+5ο…οŠ¨
  • Bπ‘˜(π‘₯)=(π‘₯+2)(3π‘₯βˆ’5)ο€Ήπ‘₯βˆ’2π‘₯+5ο…οŠ¨
  • Cπ‘˜(π‘₯)=βˆ’(π‘₯βˆ’2)(3π‘₯+5)ο€Ήπ‘₯βˆ’2π‘₯+5ο…οŠ¨
  • Dπ‘˜(π‘₯)=(π‘₯+2)(3π‘₯βˆ’5)ο€Ήπ‘₯+2π‘₯+5ο…οŠ¨
  • Eπ‘˜(π‘₯)=βˆ’(π‘₯+2)(3π‘₯βˆ’5)ο€Ήπ‘₯+2π‘₯+5ο…οŠ¨

Write π‘˜(π‘₯) as the product of linear factors.

  • Aπ‘˜(π‘₯)=βˆ’(π‘₯βˆ’2)(3π‘₯+5)(π‘₯+1βˆ’2𝑖)(π‘₯+1+2𝑖)
  • Bπ‘˜(π‘₯)=(π‘₯+2)(3π‘₯βˆ’5)(π‘₯βˆ’1βˆ’2𝑖)(π‘₯βˆ’1+2𝑖)
  • Cπ‘˜(π‘₯)=βˆ’(π‘₯βˆ’2)(3π‘₯+5)(π‘₯βˆ’1βˆ’2𝑖)(π‘₯βˆ’1+2𝑖)
  • Dπ‘˜(π‘₯)=(π‘₯+2)(3π‘₯βˆ’5)(π‘₯+1βˆ’2𝑖)(π‘₯+1+2𝑖)
  • Eπ‘˜(π‘₯)=βˆ’(π‘₯+2)(3π‘₯βˆ’5)(π‘₯+1βˆ’2𝑖)(π‘₯+1+2𝑖)

List all zeros of π‘˜(π‘₯).

  • A2,βˆ’53,1+2𝑖,1βˆ’2𝑖
  • Bβˆ’2,53,1+2𝑖,1βˆ’2𝑖
  • C2,βˆ’53,βˆ’1+2𝑖,βˆ’1βˆ’2𝑖
  • Dβˆ’2,53,βˆ’1+2𝑖,βˆ’1βˆ’2𝑖

Q8:

If 𝑓(π‘₯) is an irreducible polynomial, with 𝑓(π‘₯)βˆ£π‘(π‘₯)π‘ž(π‘₯), then which of the following is true?

  • A𝑓(π‘₯)βˆ£π‘(π‘₯) or 𝑓(π‘₯)βˆ£π‘ž(π‘₯)
  • B𝑓(π‘₯)=𝑝(π‘₯)π‘ž(π‘₯).
  • C𝑓(π‘₯) is a zero divisor.
  • D𝑓(π‘₯) = either 𝑝(π‘₯) or π‘ž(π‘₯).

Q9:

What is the least possible degree of a polynomial function with rational coefficients, given that it has βˆ’1 (multiplicity 2), 1+8𝑖, and 8βˆ’βˆš7 (multiplicity 3) as zeros?

Q10:

What is the least possible degree of a polynomial function with rational coefficients, given that it has βˆ’6, 2βˆ’βˆš7, and 6+2𝑖 as zeros?

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