Lesson Worksheet: Roots of Cubic Functions Mathematics • 10th Grade

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


Find the set of zeros of the function 𝑓(π‘₯)=7π‘₯(π‘₯βˆ’1)(π‘₯+6).

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


Find the set of zeros of the function 𝑓(π‘₯)=π‘₯+5π‘₯βˆ’9π‘₯βˆ’45.

  • A{βˆ’5,3}
  • B{5,3,βˆ’3}
  • C{βˆ’5,βˆ’3}
  • D{βˆ’5,3,βˆ’3}
  • E{5,3}


Find the value of π‘Ž, given the set 𝑧(𝑓)={βˆ’2} contains the zero of the function 𝑓(π‘₯)=π‘₯βˆ’π‘₯+π‘ŽοŠ©οŠ¨.


Solve the equation (3π‘₯βˆ’2)(5π‘₯+2)(7π‘₯βˆ’3)=0.

  • Aπ‘₯=βˆ’23, π‘₯=25, π‘₯=βˆ’37
  • Bπ‘₯=βˆ’2, π‘₯=2, π‘₯=βˆ’3
  • Cπ‘₯=32, π‘₯=βˆ’52, π‘₯=73
  • Dπ‘₯=2, π‘₯=βˆ’2, π‘₯=3
  • Eπ‘₯=23, π‘₯=βˆ’25, π‘₯=37


Solve the equation (π‘₯βˆ’2)(π‘₯+2)(π‘₯βˆ’3)=0.

  • Aπ‘₯=2, π‘₯=2, π‘₯=3
  • Bπ‘₯=2, π‘₯=2, π‘₯=βˆ’3
  • Cπ‘₯=βˆ’2, π‘₯=2, π‘₯=βˆ’3
  • Dπ‘₯=2, π‘₯=βˆ’2, π‘₯=3
  • Eπ‘₯=βˆ’2, π‘₯=βˆ’2, π‘₯=βˆ’3


Determine the solution set of the equation ο€Ήπ‘¦βˆ’72=βˆ’512 in β„€.

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


Solve π‘₯=8.

  • Aπ‘₯=2
  • Bπ‘₯=3
  • Cπ‘₯=24
  • Dπ‘₯=2 or π‘₯=βˆ’2
  • Eπ‘₯=24 or π‘₯=βˆ’24


Solve π‘₯+10=74.

  • Aπ‘₯=8
  • Bπ‘₯=8 or π‘₯=βˆ’8
  • Cπ‘₯=4
  • Dπ‘₯=9
  • Eπ‘₯=4 or π‘₯=βˆ’4


Find the solution set of 81π‘₯=121π‘₯ in ℝ.

  • A911,βˆ’911
  • B119
  • C0,119,βˆ’119
  • D0,911,βˆ’911
  • E119,βˆ’119


The figure shows the curve 𝑦=π‘₯βˆ’2π‘₯ together with the line 𝑦=π‘˜(π‘₯βˆ’1)βˆ’1 which has slope π‘˜ and passes through point (1,βˆ’1).

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

  • Aπ‘₯+(π‘˜+2)π‘₯+π‘˜+1
  • Bπ‘₯βˆ’(π‘˜+2)π‘₯+π‘˜βˆ’1
  • Cπ‘₯βˆ’(π‘˜+2)π‘₯+π‘˜+1
  • Dπ‘₯βˆ’(π‘˜βˆ’2)π‘₯+π‘˜+1
  • Eπ‘₯βˆ’2π‘₯

Divide this polynomial by π‘₯βˆ’1 to get a quadratic that is a multiple of (π‘₯βˆ’π‘Ž)(π‘₯βˆ’π‘).

  • Aπ‘₯+π‘₯+π‘˜+1
  • Bπ‘₯+π‘₯βˆ’π‘˜βˆ’1
  • Cπ‘₯βˆ’π‘₯βˆ’π‘˜βˆ’1
  • Dπ‘₯+π‘₯βˆ’π‘˜οŠ¨
  • Eπ‘₯+π‘₯+π‘˜βˆ’1

Since 𝑏>π‘Ž, determine 𝑏 in terms of π‘˜.

  • Aβˆšπ‘˜+1
  • Bβˆ’1+√4π‘˜+52
  • Cβˆ’1βˆ’βˆš4π‘˜+52
  • D1π‘˜+1
  • Eβˆ’1+√4π‘˜+12

Imagine changing the value of the slope π‘˜ 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 (βˆ’1,1). Determine the equation of the tangent to the curve at the point (βˆ’1,1).

  • A𝑦=π‘₯+2
  • B𝑦=π‘₯βˆ’2
  • C𝑦=π‘₯
  • D𝑦=5π‘₯βˆ’6
  • E𝑦=3π‘₯βˆ’4

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