Lesson Worksheet: Roots of Cubic Functions Mathematics

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


Find the set of zeros of the function 𝑓(π‘₯)=π‘₯(π‘₯βˆ’2)(π‘₯βˆ’7).

  • A{0,βˆ’7,βˆ’2}
  • B{7,2}
  • C{βˆ’1,7,2}
  • D{0,7,2}
  • E0,7,12


Find the set of zeros of the function 𝑓(π‘₯)=π‘₯βˆ’4π‘₯βˆ’9π‘₯+36.

  • A{4,3}
  • B{βˆ’4,3,βˆ’3}
  • C{4,βˆ’3}
  • D{4,3,βˆ’3}
  • E{βˆ’4,3}


Find the set of zeros of the function 𝑓(π‘₯)=π‘₯βˆ’2π‘₯βˆ’6π‘₯+27.

  • A{3}
  • B{0}
  • Cβˆ…
  • D{βˆ’3}
  • E{0,βˆ’3}


Find the set of zeros of the function 𝑓(π‘₯)=π‘₯βˆ’2π‘₯βˆ’16π‘₯+32.

  • A{2,4}
  • B{βˆ’2,4,βˆ’4}
  • C{2,βˆ’4}
  • D{2,4,βˆ’4}
  • E{βˆ’2,4}


Find the set of zeros of the function 𝑓(π‘₯)=π‘₯βˆ’2π‘₯βˆ’25π‘₯+50.

  • A{2,5}
  • B{βˆ’2,5,βˆ’5}
  • C{2,βˆ’5}
  • D{2,5,βˆ’5}
  • E{βˆ’2,5}


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


Liam has listed what he thinks are the possible types of real roots for a cubic function:

  1. One real root,
  2. Three equal real roots,
  3. Three real roots where two are equal and one is distinct,
  4. Three distinct real roots.

Sophia says that it’s possible to have exactly two real roots, which are distinct. Is Sophia correct?

  • AYes
  • BNo

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.