Worksheet: Roots of Cubic Functions

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In this worksheet, we will practice finding the roots of cubic functions with integer coefficients.

Q1:

Find the set of zeros of the function 𝑓(𝑥)=7𝑥(𝑥−1)(𝑥+6).

  • A { 0 , 6 , − 1 }
  • B { − 6 , 1 }
  • C { − 7 , − 6 , 1 }
  • D { 0 , − 6 , 1 }

Q2:

Find the set of zeros of the function 𝑓(𝑥)=𝑥+5𝑥−9𝑥−45.

  • A { − 5 , 3 }
  • B { 5 , 3 , − 3 }
  • C { − 5 , − 3 }
  • D { − 5 , 3 , − 3 }
  • E { 5 , 3 }

Q3:

Find the value of 𝑎, given the set 𝑧(𝑓)={−2} contains the zero of the function 𝑓(𝑥)=𝑥−𝑥+𝑎.

Q4:

Solve the equation (3𝑥−2)(5𝑥+2)(7𝑥−3)=0.

  • A 𝑥 = − 2 3 , 𝑥 = 2 5 , 𝑥 = − 3 7
  • B 𝑥 = − 2 , 𝑥 = 2 , 𝑥 = − 3
  • C 𝑥 = 3 2 , 𝑥 = − 5 2 , 𝑥 = 7 3
  • D 𝑥 = 2 , 𝑥 = − 2 , 𝑥 = 3
  • E 𝑥 = 2 3 , 𝑥 = − 2 5 , 𝑥 = 3 7

Q5:

Solve the equation (𝑥−2)(𝑥+2)(𝑥−3)=0.

  • A 𝑥 = 2 , 𝑥 = 2 , 𝑥 = 3
  • B 𝑥 = 2 , 𝑥 = 2 , 𝑥 = − 3
  • C 𝑥 = − 2 , 𝑥 = 2 , 𝑥 = − 3
  • D 𝑥 = 2 , 𝑥 = − 2 , 𝑥 = 3
  • E 𝑥 = − 2 , 𝑥 = − 2 , 𝑥 = − 3

Q6:

Determine the solution set of the equation 𝑦−72=−512 in ℤ.

  • A { − 8 0 }
  • B { − 8 , 8 }
  • C { 6 4 }
  • D { 8 }
  • E { − 8 }

Q7:

Solve 𝑥=8.

  • A 𝑥 = 2
  • B 𝑥 = 3
  • C 𝑥 = 2 4
  • D 𝑥 = 2 or 𝑥=−2
  • E 𝑥 = 2 4 or 𝑥=−24

Q8:

Solve 𝑥+10=74.

  • A 𝑥 = 8
  • B 𝑥 = 8 or 𝑥=−8
  • C 𝑥 = 4
  • D 𝑥 = 9
  • E 𝑥 = 4 or 𝑥=−4

Q9:

Find the solution set of 81𝑥=121𝑥 in ℝ.

  • A 9 1 1 , − 9 1 1
  • B 1 1 9
  • C 0 , 1 1 9 , − 1 1 9
  • D 0 , 9 1 1 , − 9 1 1
  • E 1 1 9 , − 1 1 9

Q10:

Find the set of zeros of the function 𝑓(𝑥)=𝑥(𝑥−2)(𝑥−7).

  • A { 0 , − 7 , − 2 }
  • B { 7 , 2 }
  • C { − 1 , 7 , 2 }
  • D { 0 , 7 , 2 }
  • E 0 , 7 , 1 2

Q11:

Find the set of zeros of the function 𝑓(𝑥)=𝑥−4𝑥−9𝑥+36.

  • A { 4 , 3 }
  • B { − 4 , 3 , − 3 }
  • C { 4 , − 3 }
  • D { 4 , 3 , − 3 }
  • E { − 4 , 3 }

Q12:

Find the set of zeros of the function 𝑓(𝑥)=𝑥−2𝑥−6𝑥+27.

  • A { 3 }
  • B { 0 }
  • C ∅
  • D { − 3 }
  • E { 0 , − 3 }

Q13:

Find the set of zeros of the function 𝑓(𝑥)=𝑥−2𝑥−16𝑥+32.

  • A { 2 , 4 }
  • B { − 2 , 4 , − 4 }
  • C { 2 , − 4 }
  • D { 2 , 4 , − 4 }
  • E { − 2 , 4 }

Q14:

Find the set of zeros of the function 𝑓(𝑥)=𝑥−2𝑥−25𝑥+50.

  • A { 2 , 5 }
  • B { − 2 , 5 , − 5 }
  • C { 2 , − 5 }
  • D { 2 , 5 , − 5 }
  • E { − 2 , 5 }

Q15:

The figure shows the curve 𝑦=𝑥−2𝑥 together with the line 𝑦=𝑘(𝑥−1)−1 which has slope 𝑘 and passes through point (1,−1).

Write a cubic polynomial whose roots are 𝑔, ℎ, and 1.

  • A 𝑥 + ( 𝑘 + 2 ) 𝑥 + 𝑘 + 1
  • B 𝑥 − ( 𝑘 + 2 ) 𝑥 + 𝑘 − 1
  • C 𝑥 − ( 𝑘 + 2 ) 𝑥 + 𝑘 + 1
  • D 𝑥 − ( 𝑘 − 2 ) 𝑥 + 𝑘 + 1
  • E 𝑥 − 2 𝑥

Divide this polynomial by 𝑥−1 to get a quadratic that is a multiple of (𝑥−𝑎)(𝑥−𝑏).

  • A 𝑥 + 𝑥 + 𝑘 + 1
  • B 𝑥 + 𝑥 − 𝑘 − 1
  • C 𝑥 − 𝑥 − 𝑘 − 1
  • D 𝑥 + 𝑥 − 𝑘
  • E 𝑥 + 𝑥 + 𝑘 − 1

Since 𝑏>𝑎, determine 𝑏 in terms of 𝑘.

  • A √ 𝑘 + 1
  • B − 1 + √ 4 𝑘 + 5 2
  • C − 1 − √ 4 𝑘 + 5 2
  • D 1 𝑘 + 1
  • E − 1 + √ 4 𝑘 + 1 2

Imagine changing the value of the slope 𝑘 so that the value of 𝑏 gets closer and closer to 1. When 𝑏=1, the line will be tangent to the curve at the point (−1,1). Determine the equation of the tangent to the curve at the point (−1,1).

  • A 𝑦 = 𝑥 + 2
  • B 𝑦 = 𝑥 − 2
  • C 𝑦 = 𝑥
  • D 𝑦 = 5 𝑥 − 6
  • E 𝑦 = 3 𝑥 − 4

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