Lesson Worksheet: Linear Factorization and Conjugate Root Theorems Mathematics

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)=𝑥17𝑥+93𝑥131𝑥106
  • B𝑓(𝑥)=𝑥17𝑥9𝑥+131𝑥106
  • C𝑓(𝑥)=𝑥+17𝑥+97𝑥+187𝑥+106
  • D𝑓(𝑥)=𝑥17𝑥+97𝑥187𝑥+106
  • E𝑓(𝑥)=𝑥+17𝑥13𝑥187𝑥+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)=𝑥4𝑥8𝑥+32𝑥+16𝑥64
  • C𝑓(𝑥)=𝑥+4𝑥8𝑥32𝑥+16𝑥+64
  • D𝑓(𝑥)=𝑥4𝑥+8𝑥32𝑥+16𝑥64

Q3:

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

  • A𝑓(𝑥)=𝑥10𝑥+35𝑥46𝑥+10
  • B𝑓(𝑥)=𝑥10𝑥+13𝑥+46𝑥+10
  • C𝑓(𝑥)=𝑥10𝑥+33𝑥34𝑥10
  • D𝑓(𝑥)=𝑥2𝑥15𝑥+46𝑥10
  • E𝑓(𝑥)=𝑥+10𝑥+33𝑥+34𝑥10

Q4:

Consider 𝑔(𝑥)=𝑥7𝑥+11𝑥41𝑥+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)=(𝑥4)(𝑥5)(𝑥+𝑥+4)
  • D𝑔(𝑥)=(𝑥4)(𝑥5)(𝑥+2𝑥+9)

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

  • A𝑔(𝑥)=(𝑥+4)(𝑥+5)𝑥+122𝑖𝑥+1+22𝑖
  • B𝑔(𝑥)=(𝑥+4)(𝑥+5)𝑥+12152𝑖𝑥+12+152𝑖
  • C𝑔(𝑥)=(𝑥4)(𝑥5)𝑥+122𝑖𝑥+1+22𝑖
  • D𝑔(𝑥)=(𝑥4)(𝑥5)𝑥+12152𝑖𝑥+12+152𝑖

List all zeros of 𝑔(𝑥).

  • A5,4,12152𝑖,12+152𝑖
  • B4,5,12152𝑖,12+152𝑖
  • C4,5,1+22𝑖,122𝑖
  • D5,4,1+22𝑖,122𝑖

Q5:

Consider (𝑥)=5𝑥4𝑥81𝑥+134𝑥+30.

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

  • A(𝑥)=(𝑥3)(5𝑥+1)(𝑥+2𝑥10)
  • B(𝑥)=(𝑥+3)(5𝑥1)(𝑥+1+11)(𝑥+111)
  • C(𝑥)=(𝑥3)(5𝑥+1)(𝑥1+11)(𝑥111)
  • D(𝑥)=(𝑥3)(5𝑥+1)(𝑥+1+11)(𝑥+111)
  • E(𝑥)=(𝑥+3)(5𝑥1)(𝑥+2𝑥10)

List all zeros of (𝑥).

  • A3,15
  • B3,15
  • C3,15,111,111
  • D3,15,111,111
  • E3,15,111,1+11

Q6:

Consider 𝑓(𝑥)=𝑥+3𝑥5𝑥3𝑥+4.

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

  • A𝑓(𝑥)=(𝑥1)(𝑥4)(𝑥+1)
  • B𝑓(𝑥)=(𝑥+4)(𝑥1)
  • C𝑓(𝑥)=(𝑥+4)(𝑥+1)
  • D𝑓(𝑥)=(𝑥1)(𝑥+4)(𝑥+1)
  • E𝑓(𝑥)=(𝑥+1)(𝑥+4)(𝑥1)

List all zeros of 𝑓(𝑥).

  • A1,4,1
  • B4,1
  • C1,4,1
  • D4,1

Q7:

Consider 𝑘(𝑥)=3𝑥7𝑥7𝑥+15𝑥+50.

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

  • A𝑘(𝑥)=(𝑥2)(3𝑥+5)𝑥+2𝑥+5
  • B𝑘(𝑥)=(𝑥+2)(3𝑥5)𝑥2𝑥+5
  • C𝑘(𝑥)=(𝑥2)(3𝑥+5)𝑥2𝑥+5
  • D𝑘(𝑥)=(𝑥+2)(3𝑥5)𝑥+2𝑥+5
  • E𝑘(𝑥)=(𝑥+2)(3𝑥5)𝑥+2𝑥+5

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

  • A𝑘(𝑥)=(𝑥2)(3𝑥+5)(𝑥+12𝑖)(𝑥+1+2𝑖)
  • B𝑘(𝑥)=(𝑥+2)(3𝑥5)(𝑥12𝑖)(𝑥1+2𝑖)
  • C𝑘(𝑥)=(𝑥2)(3𝑥+5)(𝑥12𝑖)(𝑥1+2𝑖)
  • D𝑘(𝑥)=(𝑥+2)(3𝑥5)(𝑥+12𝑖)(𝑥+1+2𝑖)
  • E𝑘(𝑥)=(𝑥+2)(3𝑥5)(𝑥+12𝑖)(𝑥+1+2𝑖)

List all zeros of 𝑘(𝑥).

  • A2,53,1+2𝑖,12𝑖
  • B2,53,1+2𝑖,12𝑖
  • C2,53,1+2𝑖,12𝑖
  • D2,53,1+2𝑖,12𝑖

Q8:

What is the least possible degree of a polynomial function with rational coefficients, given that it has 1 (multiplicity 2), 1+8𝑖, and 87 (multiplicity 3) as zeros?

Q9:

What is the least possible degree of a polynomial function with rational coefficients, given that it has 6, 27, and 6+2𝑖 as zeros?

This lesson includes 2 additional questions and 18 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.