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Lesson Worksheet: Factor Theorem Mathematics

In this worksheet, we will practice using the factor theorem to factor polynomials, especially cubic polynomials.

Q1:

Using the factor theorem, determine which of the following linear expressions is a factor of 6𝑥+17𝑥31𝑥12.

  • A(𝑥3)
  • B(𝑥+4)
  • C(𝑥+6)
  • D(𝑥+3)
  • E(𝑥4)

Q2:

Given that 𝑓(1)=0 and 𝑓(3)=0, fully factor 𝑓(𝑥)=8𝑥+30𝑥+7𝑥36𝑥9.

  • A(𝑥+1)(𝑥+3)(2𝑥+3)(4𝑥+1)
  • B(𝑥+1)(𝑥3)(2𝑥+3)(4𝑥+1)
  • C(𝑥1)(𝑥+3)(𝑥+3)(𝑥+1)
  • D(𝑥1)(𝑥+3)(2𝑥+3)(4𝑥+1)
  • E(𝑥+1)(𝑥3)(𝑥+3)(𝑥+1)

Q3:

For a polynomial expression 𝑓(𝑥), if 𝑓(3)=0, which of the following can be said to be a factor of 𝑓(𝑥) using the factor theorem?

  • A𝑥+3
  • B(𝑥+3)
  • CWe cannot determine any factor of 𝑓(𝑥).
  • D𝑥3
  • E(𝑥3)

Q4:

Given that (𝑥3) is a factor of 2𝑥6𝑥+5𝑥+𝑎, what is the value of 𝑎?

Q5:

Given that 𝑥+3𝑥33𝑥35 has a factor (𝑥𝑝), where 𝑝 is an integer and |𝑝|2, find 𝑝.

Hence, fully factor 𝑥+3𝑥33𝑥35.

  • A(𝑥7)(𝑥1)(𝑥+5)
  • B(𝑥+7)(𝑥1)(𝑥5)
  • C(𝑥+7)(𝑥+1)(𝑥5)
  • D(𝑥7)(𝑥1)(𝑥5)
  • E(𝑥+7)(𝑥+1)(𝑥+5)

Q6:

Using the factor theorem, determine which of the following linear expressions is a factor of 𝑥+𝑥17𝑥+15.

  • A(𝑥2)
  • B(𝑥+1)
  • C(𝑥1)
  • D(𝑥5)
  • E(𝑥+3)

Q7:

Using the factor theorem, factor 6𝑥+13𝑥+𝑥2 completely.

  • A(3𝑥+1)(2𝑥+1)(𝑥+2)
  • B(3𝑥1)(2𝑥+1)(𝑥+2)
  • C(3𝑥+1)(2𝑥1)(𝑥2)
  • D(𝑥1)(𝑥1)(𝑥2)
  • E(𝑥1)(𝑥+1)(𝑥+2)

Q8:

Suppose that (𝑥+4) is a factor of 𝑥+𝑎𝑥8𝑥. What is the value of 𝑎?

Q9:

For a polynomial expression 𝑓(𝑥), if 𝑓(2)=1, is (𝑥2) a factor of 𝑓(𝑥)?

  • AMore information is needed.
  • BYes
  • CNo

Q10:

What is the completely factored form of 15𝑥14𝑥43𝑥14?

  • A(𝑥+2)(𝑥+7)(𝑥+1)
  • B(𝑥+2)(𝑥7)(𝑥+1)
  • C(5𝑥+2)(3𝑥+7)(𝑥+1)
  • D(5𝑥2)(3𝑥+7)(𝑥1)
  • E(5𝑥+2)(3𝑥7)(𝑥+1)

Using this factored form, identify the graph of 𝑦=15𝑥14𝑥43𝑥14.

  • A
  • B
  • C
  • D
  • E

This lesson includes 90 additional question variations for subscribers.

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