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.

  • A𝑥=−23, 𝑥=25, 𝑥=−37
  • B𝑥=−2, 𝑥=2, 𝑥=−3
  • C𝑥=32, 𝑥=−52, 𝑥=73
  • D𝑥=2, 𝑥=−2, 𝑥=3
  • E𝑥=23, 𝑥=−25, 𝑥=37

Q5:

Solve the equation (𝑥−2)(𝑥+2)(𝑥−3)=0.

  • A𝑥=2, 𝑥=2, 𝑥=3
  • B𝑥=2, 𝑥=2, 𝑥=−3
  • C𝑥=−2, 𝑥=2, 𝑥=−3
  • D𝑥=2, 𝑥=−2, 𝑥=3
  • E𝑥=−2, 𝑥=−2, 𝑥=−3

Q6:

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

  • A{−80}
  • B{−8,8}
  • C{64}
  • D{8}
  • E{−8}

Q7:

Solve 𝑥=8.

  • A𝑥=2
  • B𝑥=3
  • C𝑥=24
  • D𝑥=2 or 𝑥=−2
  • E𝑥=24 or 𝑥=−24

Q8:

Solve 𝑥+10=74.

  • A𝑥=8
  • B𝑥=8 or 𝑥=−8
  • C𝑥=4
  • D𝑥=9
  • E𝑥=4 or 𝑥=−4

Q9:

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

  • A911,−911
  • B119
  • C0,119,−119
  • D0,911,−911
  • E119,−119

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}
  • E0,7,12

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.

  • A𝑥+(𝑘+2)𝑥+𝑘+1
  • B𝑥−(𝑘+2)𝑥+𝑘−1
  • C𝑥−(𝑘+2)𝑥+𝑘+1
  • D𝑥−(𝑘−2)𝑥+𝑘+1
  • E𝑥−2𝑥

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

  • A𝑥+𝑥+𝑘+1
  • B𝑥+𝑥−𝑘−1
  • C𝑥−𝑥−𝑘−1
  • D𝑥+𝑥−𝑘
  • E𝑥+𝑥+𝑘−1

Since 𝑏>𝑎, determine 𝑏 in terms of 𝑘.

  • A√𝑘+1
  • B−1+√4𝑘+52
  • C−1−√4𝑘+52
  • D1𝑘+1
  • E−1+√4𝑘+12

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).

  • A𝑦=𝑥+2
  • B𝑦=𝑥−2
  • C𝑦=𝑥
  • D𝑦=5𝑥−6
  • E𝑦=3𝑥−4

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