Worksheet: Roots of Cubic Functions

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(𝑥1)(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+5𝑥9𝑥45.

  • 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)=𝑥𝑥+𝑎.

Q4:

Solve the equation (3𝑥2)(5𝑥+2)(7𝑥3)=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 (𝑥2)(𝑥+2)(𝑥3)=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 𝑦72=512 in .

  • 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 .

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

Q10:

Find the set of zeros of the function 𝑓(𝑥)=𝑥(𝑥2)(𝑥7).

  • 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)=𝑥4𝑥9𝑥+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)=𝑥2𝑥6𝑥+27.

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

Q13:

Find the set of zeros of the function 𝑓(𝑥)=𝑥2𝑥16𝑥+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)=𝑥2𝑥25𝑥+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=𝑥2𝑥 together with the line 𝑦=𝑘(𝑥1)1 which has slope 𝑘 and passes through point (1,1).

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

  • A𝑥+(𝑘+2)𝑥+𝑘+1
  • B𝑥(𝑘+2)𝑥+𝑘1
  • C𝑥(𝑘+2)𝑥+𝑘+1
  • D𝑥(𝑘2)𝑥+𝑘+1
  • E𝑥2𝑥

Divide this polynomial by 𝑥1 to get a quadratic that is a multiple of (𝑥𝑎)(𝑥𝑏).

  • A𝑥+𝑥+𝑘+1
  • B𝑥+𝑥𝑘1
  • C𝑥𝑥𝑘1
  • D𝑥+𝑥𝑘
  • E𝑥+𝑥+𝑘1

Since 𝑏>𝑎, determine 𝑏 in terms of 𝑘.

  • A𝑘+1
  • B1+4𝑘+52
  • C14𝑘+52
  • D1𝑘+1
  • E1+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=𝑥2
  • C𝑦=𝑥
  • D𝑦=5𝑥6
  • E𝑦=3𝑥4

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