Lesson Worksheet: Polynomial Long Division with Remainder Mathematics • 10th Grade

In this worksheet, we will practice finding the quotient and remainder when polynomials are divided, including the case when the divisor is irreducible.

Q1:

Use polynomial division to simplify 2π‘₯+5π‘₯+7π‘₯+4π‘₯+1.

  • Aπ‘₯+5π‘₯+2
  • Bπ‘₯+3π‘₯+4
  • C2π‘₯+5π‘₯+4
  • D2π‘₯+3π‘₯+4
  • E2π‘₯+5π‘₯+2

Q2:

Write π‘₯βˆ’2π‘₯βˆ’21π‘₯βˆ’7π‘₯+6π‘₯+3π‘₯βˆ’2οŠͺ in the form π‘ž(π‘₯)+π‘Ÿ(π‘₯)𝑑(π‘₯) where the degree of π‘Ÿ(π‘₯) is less than that of 𝑑(π‘₯).

  • Aπ‘₯+π‘₯βˆ’20βˆ’65π‘₯βˆ’34π‘₯+3π‘₯βˆ’2
  • Bπ‘₯+π‘₯βˆ’20+65π‘₯βˆ’34π‘₯+3π‘₯βˆ’2
  • Cπ‘₯βˆ’5π‘₯βˆ’4βˆ’5π‘₯+2π‘₯+3π‘₯βˆ’2
  • Dπ‘₯βˆ’5π‘₯βˆ’4βˆ’π‘₯+3π‘₯βˆ’25π‘₯+2
  • Eπ‘₯βˆ’5π‘₯βˆ’4+5π‘₯+2π‘₯+3π‘₯βˆ’2

Q3:

Find the remainder π‘Ÿ(π‘₯), and the quotient π‘ž(π‘₯) when 4π‘₯+2π‘₯βˆ’π‘₯βˆ’6οŠͺ is divided by 2π‘₯βˆ’4π‘₯+1.

  • Aπ‘Ÿ(π‘₯)=30π‘₯βˆ’15,π‘ž(π‘₯)=2π‘₯+5π‘₯+9
  • Bπ‘Ÿ(π‘₯)=30π‘₯βˆ’13,π‘ž(π‘₯)=2π‘₯βˆ’3π‘₯+7
  • Cπ‘Ÿ(π‘₯)=βˆ’5π‘₯βˆ’5,π‘ž(π‘₯)=2π‘₯+1
  • Dπ‘Ÿ(π‘₯)=38π‘₯βˆ’17,π‘ž(π‘₯)=2π‘₯+5π‘₯+11
  • Eπ‘Ÿ(π‘₯)=30π‘₯βˆ’15,π‘ž(π‘₯)=2π‘₯βˆ’4π‘₯+1

Q4:

Write π‘₯βˆ’2π‘₯βˆ’17π‘₯βˆ’3π‘₯+4π‘₯βˆ’2π‘₯+3π‘₯οŠͺ in the form π‘ž(π‘₯)+π‘Ÿ(π‘₯)𝑑(π‘₯).

  • Aπ‘₯βˆ’5π‘₯βˆ’2π‘₯+3+5π‘₯+2π‘₯+3π‘₯
  • Bπ‘₯+π‘₯βˆ’14π‘₯+45βˆ’π‘₯+3π‘₯4π‘₯βˆ’2
  • Cπ‘₯βˆ’5π‘₯βˆ’2π‘₯+3βˆ’5π‘₯+2π‘₯+3π‘₯
  • Dπ‘₯+π‘₯βˆ’14π‘₯+45+4π‘₯βˆ’2π‘₯+3π‘₯
  • Eπ‘₯βˆ’5π‘₯βˆ’2π‘₯+3βˆ’π‘₯+3π‘₯5π‘₯+2

Q5:

Given that π‘₯+4π‘₯βˆ’2π‘₯βˆ’3=π‘₯+7 with a remainder of 19, rewrite π‘₯+4π‘₯βˆ’2 in the form (π‘₯βˆ’π‘Ž)Γ—π‘ž(π‘₯)+𝑓(π‘Ž).

  • Aπ‘₯+4π‘₯βˆ’2=(π‘₯+2)(π‘₯βˆ’1)+19
  • Bπ‘₯+4π‘₯βˆ’2=(π‘₯+7)(π‘₯βˆ’3)+19
  • Cπ‘₯+4π‘₯βˆ’2=(π‘₯βˆ’7)(π‘₯+3)+19
  • Dπ‘₯+4π‘₯βˆ’2=(π‘₯βˆ’7)(π‘₯βˆ’3)+19
  • Eπ‘₯+4π‘₯βˆ’2=(π‘₯+7)(π‘₯+3)+19

Q6:

Find the remainder when 3π‘₯βˆ’2π‘₯+4π‘₯+5 is divided by 3π‘₯+4.

Q7:

Find the remainder when 5π‘₯+2π‘₯βˆ’8 is divided by π‘₯βˆ’2.

Q8:

Write 3π‘₯+4π‘₯+5π‘₯+10π‘₯+5 in the form of π‘ž(π‘₯)+π‘Ÿ(π‘₯)𝑑(π‘₯).

  • A3π‘₯βˆ’11π‘₯+60+290π‘₯+5
  • B3π‘₯+19π‘₯βˆ’90+460π‘₯+5
  • C3π‘₯+19π‘₯βˆ’90+π‘₯+5460
  • D3π‘₯βˆ’11π‘₯+60βˆ’π‘₯+5290
  • E3π‘₯βˆ’11π‘₯+60βˆ’290π‘₯+5

Q9:

Find the remainder when 2π‘₯+3π‘₯+2 is divided by π‘₯+1.

Q10:

Write 3π‘₯+4π‘₯+13π‘₯+2 in the form π‘ž(π‘₯)+π‘Ÿ(π‘₯)𝑑(π‘₯).

  • A3π‘₯+10π‘₯+20+π‘₯+253
  • Bπ‘₯βˆ’2π‘₯+4+5π‘₯+2
  • C3π‘₯+10π‘₯+20+53π‘₯+2
  • D3π‘₯βˆ’2π‘₯+4+π‘₯+25
  • E3π‘₯βˆ’2π‘₯+4+5π‘₯+2

This lesson includes 9 additional questions for subscribers.

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