Lesson Worksheet: Linear Factorization and Conjugate Root Theorems Mathematics • 10th Grade

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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(𝑥)=𝑥−17𝑥+93𝑥−131𝑥−106
  • B𝑓(𝑥)=𝑥−17𝑥−9𝑥+131𝑥−106
  • C𝑓(𝑥)=𝑥+17𝑥+97𝑥+187𝑥+106
  • D𝑓(𝑥)=𝑥−17𝑥+97𝑥−187𝑥+106
  • E𝑓(𝑥)=𝑥+17𝑥−13𝑥−187𝑥+106

Q2:

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

  • A𝑓(𝑥)=𝑥+4𝑥+8𝑥+32𝑥+16𝑥+64
  • B𝑓(𝑥)=𝑥−4𝑥−8𝑥+32𝑥+16𝑥−64
  • C𝑓(𝑥)=𝑥+4𝑥−8𝑥−32𝑥+16𝑥+64
  • D𝑓(𝑥)=𝑥−4𝑥+8𝑥−32𝑥+16𝑥−64

Q3:

Write a polynomial function of the least degree with real coefficients in standard form given that it has √5+2, −√5+2 and 3−𝑖 as zeros.

  • A𝑓(𝑥)=𝑥−10𝑥+35𝑥−46𝑥+10
  • B𝑓(𝑥)=𝑥−10𝑥+13𝑥+46𝑥+10
  • C𝑓(𝑥)=𝑥−10𝑥+33𝑥−34𝑥−10
  • D𝑓(𝑥)=𝑥−2𝑥−15𝑥+46𝑥−10
  • E𝑓(𝑥)=𝑥+10𝑥+33𝑥+34𝑥−10

Q4:

Consider 𝑔(𝑥)=𝑥−7𝑥+11𝑥−41𝑥+180.

Write 𝑔(𝑥) as the product of linear and irreducible quadratic factors.

  • A𝑔(𝑥)=(𝑥+4)(𝑥+5)(𝑥+𝑥+4)
  • B𝑔(𝑥)=(𝑥+4)(𝑥+5)(𝑥+2𝑥+9)
  • C𝑔(𝑥)=(𝑥−4)(𝑥−5)(𝑥+𝑥+4)
  • D𝑔(𝑥)=(𝑥−4)(𝑥−5)(𝑥+2𝑥+9)

Write 𝑔(𝑥) as the product of linear factors.

  • A𝑔(𝑥)=(𝑥+4)(𝑥+5)𝑥+1−2√2𝑖𝑥+1+2√2𝑖
  • B𝑔(𝑥)=(𝑥+4)(𝑥+5)𝑥+12−√152𝑖𝑥+12+√152𝑖
  • C𝑔(𝑥)=(𝑥−4)(𝑥−5)𝑥+1−2√2𝑖𝑥+1+2√2𝑖
  • D𝑔(𝑥)=(𝑥−4)(𝑥−5)𝑥+12−√152𝑖𝑥+12+√152𝑖

List all zeros of 𝑔(𝑥).

  • A−5,−4,−12−√152𝑖,−12+√152𝑖
  • B4,5,−12−√152𝑖,−12+√152𝑖
  • C4,5,−1+2√2𝑖,−1−2√2𝑖
  • D−5,−4,−1+2√2𝑖,−1−2√2𝑖

Q5:

Consider ℎ(𝑥)=5𝑥−4𝑥−81𝑥+134𝑥+30.

Write ℎ(𝑥) as the product of linear and irreducible quadratic factors.

  • Aℎ(𝑥)=(𝑥−3)(5𝑥+1)(𝑥+2𝑥−10)
  • Bℎ(𝑥)=(𝑥+3)(5𝑥−1)(𝑥+1+√11)(𝑥+1−√11)
  • Cℎ(𝑥)=(𝑥−3)(5𝑥+1)(𝑥−1+√11)(𝑥−1−√11)
  • Dℎ(𝑥)=(𝑥−3)(5𝑥+1)(𝑥+1+√11)(𝑥+1−√11)
  • Eℎ(𝑥)=(𝑥+3)(5𝑥−1)(𝑥+2𝑥−10)

List all zeros of ℎ(𝑥).

  • A3,−15
  • B−3,15
  • C−3,15,−1−√11,√11−1
  • D3,−15,−1−√11,√11−1
  • E3,−15,1−√11,1+√11

Q6:

Consider 𝑓(𝑥)=𝑥+3𝑥−5𝑥−3𝑥+4.

Write 𝑓(𝑥) as the product of linear and irreducible quadratic factors.

  • A𝑓(𝑥)=(𝑥−1)(𝑥−4)(𝑥+1)
  • B𝑓(𝑥)=(𝑥+4)(𝑥−1)
  • C𝑓(𝑥)=(𝑥+4)(𝑥+1)
  • D𝑓(𝑥)=(𝑥−1)(𝑥+4)(𝑥+1)
  • E𝑓(𝑥)=(𝑥+1)(𝑥+4)(𝑥−1)

List all zeros of 𝑓(𝑥).

  • A−1,−4,1
  • B−4,1
  • C−1,4,1
  • D−4,−1

Q7:

Consider 𝑘(𝑥)=−3𝑥−7𝑥−7𝑥+15𝑥+50.

Write 𝑘(𝑥) as the product of linear and irreducible quadratic factors.

  • A𝑘(𝑥)=−(𝑥−2)(3𝑥+5)𝑥+2𝑥+5
  • B𝑘(𝑥)=(𝑥+2)(3𝑥−5)𝑥−2𝑥+5
  • C𝑘(𝑥)=−(𝑥−2)(3𝑥+5)𝑥−2𝑥+5
  • D𝑘(𝑥)=(𝑥+2)(3𝑥−5)𝑥+2𝑥+5
  • E𝑘(𝑥)=−(𝑥+2)(3𝑥−5)𝑥+2𝑥+5

Write 𝑘(𝑥) as the product of linear factors.

  • A𝑘(𝑥)=−(𝑥−2)(3𝑥+5)(𝑥+1−2𝑖)(𝑥+1+2𝑖)
  • B𝑘(𝑥)=(𝑥+2)(3𝑥−5)(𝑥−1−2𝑖)(𝑥−1+2𝑖)
  • C𝑘(𝑥)=−(𝑥−2)(3𝑥+5)(𝑥−1−2𝑖)(𝑥−1+2𝑖)
  • D𝑘(𝑥)=(𝑥+2)(3𝑥−5)(𝑥+1−2𝑖)(𝑥+1+2𝑖)
  • E𝑘(𝑥)=−(𝑥+2)(3𝑥−5)(𝑥+1−2𝑖)(𝑥+1+2𝑖)

List all zeros of 𝑘(𝑥).

  • A2,−53,1+2𝑖,1−2𝑖
  • B−2,53,1+2𝑖,1−2𝑖
  • C2,−53,−1+2𝑖,−1−2𝑖
  • D−2,53,−1+2𝑖,−1−2𝑖

Q8:

What is the least possible degree of a polynomial function with rational coefficients, given that it has −1 (multiplicity 2), 1+8𝑖, and 8−√7 (multiplicity 3) as zeros?

Q9:

What is the least possible degree of a polynomial function with rational coefficients, given that it has −6, 2−√7, and 6+2𝑖 as zeros?

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