Worksheet: Sum and Difference of Two Cubes

In this worksheet, we will practice factoring the sum and the difference of two cubes.

Q1:

Given that ๐‘ฅโˆ’512=(๐‘ฅโˆ’8)๏€น๐‘ฅ+๐‘˜+64๏…๏Šฉ๏Šจ, find an expression for ๐‘˜.

  • A8๐‘ฅ
  • Bโˆ’8๐‘ฅ
  • Cโˆ’16๐‘ฅ
  • Dโˆ’64๐‘ฅ
  • E16๐‘ฅ

Q2:

The expression ๐‘ฅ+27๏Šฉ has two factors. One factor is (๐‘ฅ+3). What is the other factor?

  • A๏€น๐‘ฅโˆ’3๐‘ฅ+27๏…๏Šฉ
  • B๏€น๐‘ฅ+3๐‘ฅ+9๏…๏Šจ
  • C๏€น๐‘ฅ+3๐‘ฅโˆ’9๏…๏Šจ
  • D๏€น๐‘ฅโˆ’9๐‘ฅ+27๏…๏Šจ
  • E๏€น๐‘ฅโˆ’3๐‘ฅ+9๏…๏Šจ

Q3:

Factorize fully 1,000๐‘ฅโˆ’125๏Šฉ.

  • A125(2๐‘ฅโˆ’1)๏€น4๐‘ฅโˆ’4๐‘ฅ+1๏…๏Šจ
  • B125(2๐‘ฅโˆ’1)๏€น4๐‘ฅโˆ’2๐‘ฅ+1๏…๏Šจ
  • C125(2๐‘ฅโˆ’1)๏€น4๐‘ฅ+4๐‘ฅ+1๏…๏Šจ
  • D125(2๐‘ฅโˆ’1)๏€น4๐‘ฅ+2๐‘ฅ+1๏…๏Šจ

Q4:

Factorize fully 64๐‘ฅโˆ’125๐‘ฆ๏Šฉ๏Šฉ.

  • A(4๐‘ฅ+5๐‘ฆ)(4๐‘ฅโˆ’5๐‘ฆ)๏Šจ
  • B(4๐‘ฅโˆ’5๐‘ฆ)๏€น16๐‘ฅโˆ’20๐‘ฅ๐‘ฆ+25๐‘ฆ๏…๏Šจ๏Šจ
  • C(4๐‘ฅ+5๐‘ฆ)๏€น16๐‘ฅโˆ’20๐‘ฅ+25๐‘ฆ๏…๏Šจ๏Šจ
  • D(4๐‘ฅโˆ’5๐‘ฆ)๏€น16๐‘ฅ+20๐‘ฅ๐‘ฆ+25๐‘ฆ๏…๏Šจ๏Šจ

Q5:

Factorize fully 54๐‘ฅโˆ’16๐‘ฆ๏Šฉ๏Šฉ.

  • A2(3๐‘ฅโˆ’2๐‘ฆ)๏€น9๐‘ฅ+6๐‘ฅ๐‘ฆ+4๐‘ฆ๏…๏Šจ๏Šจ
  • B2(3๐‘ฅ+2๐‘ฆ)๏€น9๐‘ฅโˆ’6๐‘ฅ+4๐‘ฆ๏…๏Šจ๏Šจ
  • C2(3๐‘ฅโˆ’2๐‘ฆ)๏€น9๐‘ฅ+6๐‘ฅ๐‘ฆโˆ’4๐‘ฆ๏…๏Šจ๏Šจ
  • D(3๐‘ฅ+2๐‘ฆ)(3๐‘ฅโˆ’2๐‘ฆ)๏Šจ
  • E2(3๐‘ฅโˆ’2๐‘ฆ)๏€น9๐‘ฅโˆ’6๐‘ฅ๐‘ฆ+4๐‘ฆ๏…๏Šจ๏Šจ

Q6:

Factorize fully ๐‘ฅโˆ’18๏Šฉ.

  • A๏€ผ๐‘ฅโˆ’12๏ˆ๏€ผ๐‘ฅ+12๐‘ฅโˆ’14๏ˆ๏Šจ
  • B๏€ผ๐‘ฅโˆ’12๏ˆ๏€ผ๐‘ฅ+12๐‘ฅ+14๏ˆ๏Šจ
  • C๏€ผ๐‘ฅ+12๏ˆ๏€ผ๐‘ฅ+12๐‘ฅ+14๏ˆ๏Šจ
  • D๏€ผ๐‘ฅโˆ’12๏ˆ๏€ผ๐‘ฅโˆ’12๐‘ฅโˆ’14๏ˆ๏Šจ

Q7:

Factorize fully ๐‘Ž+๐‘๏Šจ๏Šช๏Šจ๏Šญ.

  • A๏€น๐‘Žโˆ’๐‘๏…๏€น๐‘Žโˆ’๐‘Ž๐‘+๐‘๏…๏Šฎ๏Šฏ๏Šง๏Šฌ๏Šฎ๏Šฏ๏Šง๏Šฎ
  • B๏€น๐‘Ž+๐‘๏…๏€น๐‘Ž+๐‘Ž๐‘+๐‘๏…๏Šฎ๏Šฏ๏Šง๏Šฌ๏Šฎ๏Šฏ๏Šง๏Šฎ
  • C๏€น๐‘Ž+๐‘๏…๏€น๐‘Ž+๐‘๏…๏Šฎ๏Šฏ๏Šง๏Šฌ๏Šง๏Šฎ
  • D๏€น๐‘Ž+๐‘๏…๏€น๐‘Žโˆ’๐‘Ž๐‘+๐‘๏…๏Šฎ๏Šฏ๏Šง๏Šฌ๏Šฎ๏Šฏ๏Šง๏Šฎ
  • E๏€น๐‘Žโˆ’๐‘๏…๏€น๐‘Ž+๐‘Ž๐‘+๐‘๏…๏Šฎ๏Šฏ๏Šง๏Šฌ๏Šฎ๏Šฏ๏Šง๏Šฎ

Q8:

Factorize fully ๐‘ฅ+8๐‘ฆ๏Šฉ๏Šฉ.

  • A(๐‘ฅ+2๐‘ฆ)๏€น๐‘ฅ+2๐‘ฅ๐‘ฆ+4๐‘ฆ๏…๏Šจ
  • B(๐‘ฅโˆ’2๐‘ฆ)๏€น๐‘ฅ+2๐‘ฅ๐‘ฆ+4๐‘ฆ๏…๏Šจ
  • C(๐‘ฅ+2๐‘ฆ)(๐‘ฅโˆ’2๐‘ฆ)๏Šจ
  • D(๐‘ฅ+2๐‘ฆ)๏€น๐‘ฅโˆ’2๐‘ฅ๐‘ฆ+4๐‘ฆ๏…๏Šจ๏Šจ

Q9:

Factorize fully 64๐‘ฅโˆ’729๐‘ฆ๏Šฌ๏Šฌ.

  • A(2๐‘ฅ+3๐‘ฆ)(2๐‘ฅโˆ’3๐‘ฆ)๏€น4๐‘ฅ+6๐‘ฆ๐‘ฅโˆ’9๐‘ฆ๏…๏€น4๐‘ฅโˆ’6๐‘ฆ๐‘ฅโˆ’9๐‘ฆ๏…๏Šจ๏Šจ๏Šจ๏Šจ
  • B(2๐‘ฅ+3๐‘ฆ)๏€น4๐‘ฅ+6๐‘ฆ๐‘ฅ+9๐‘ฆ๏…๏Šจ๏Šจ๏Šจ๏Šจ
  • C(2๐‘ฅ+3๐‘ฆ)(2๐‘ฅโˆ’3๐‘ฆ)๏€น4๐‘ฅ+6๐‘ฆ๐‘ฅ+9๐‘ฆ๏…๏€น4๐‘ฅโˆ’6๐‘ฆ๐‘ฅ+9๐‘ฆ๏…๏Šจ๏Šจ๏Šจ๏Šจ
  • D(2๐‘ฅ+3๐‘ฆ)๏€น4๐‘ฅ+6๐‘ฆ๐‘ฅโˆ’9๐‘ฆ๏…๏Šจ๏Šจ๏Šจ๏Šจ

Q10:

If ๐‘Žโˆ’๐‘=45๏Šฉ๏Šฉ and ๐‘Žโˆ’๐‘=9, what is the value of ๐‘Ž+๐‘Ž๐‘+๐‘๏Šจ๏Šจ?

Q11:

Complete the following: =(๐‘ฆ+15๐‘ฅ)๏€น๐‘ฆโˆ’15๐‘ฆ๐‘ฅ+225๐‘ฅ๏…๏Šจ๏Šจ.

  • A๐‘ฆโˆ’225๏Šฉ
  • B๐‘ฆ+3,375๏Šฉ
  • C๐‘ฆโˆ’15๏Šฉ
  • D๐‘ฆ+3,375๐‘ฅ๏Šฉ๏Šฉ
  • E๐‘ฆ+225๐‘ฅ๏Šฉ๏Šฉ

Q12:

Complete the factorization: 1,000๐‘ฅโˆ’729=(10๐‘ฅโˆ’9)(โ‹ฏโ‹ฏโ‹ฏโ‹ฏ)๏Šฉ.

  • A100๐‘ฅโˆ’180๐‘ฅ+81๏Šจ
  • B100๐‘ฅ+180๐‘ฅ+81๏Šจ
  • C100๐‘ฅ+90๐‘ฅ+81๏Šจ
  • Dโˆ’100๐‘ฅ+90๐‘ฅ+81๏Šจ

Q13:

Complete the following: 729๐‘ฅ+8=(+)๏€น81๐‘ฅโˆ’18๐‘ฅ+๏…๏Šฉ๏Šจ.

  • A18๐‘ฅ44,,
  • B9๐‘ฅ24,,
  • C9๐‘ฅ48,,
  • D9๐‘ฅ44,,
  • E9๐‘ฅ18,,

Q14:

If ๐‘ฅโˆ’๐‘ฆ=55๏Šจ๏Šจ, ๐‘ฅโˆ’๐‘ฆ=โˆ’5, and ๐‘ฅโˆ’๐‘ฅ๐‘ฆ+๐‘ฆ=49๏Šจ๏Šจ, what is the value of ๐‘ฅ+๐‘ฆ๏Šฉ๏Šฉ?

Q15:

What is the result of ๏€น27๐‘ฅโˆ’1๏…รท(3๐‘ฅโˆ’1)๏Šฉ, where 3๐‘ฅโ‰ 1?

  • A9๐‘ฅ+1๏Šจ
  • B(3๐‘ฅโˆ’1)๏Šจ
  • C9๐‘ฅ+3๐‘ฅ+1๏Šจ
  • D9๐‘ฅโˆ’3๐‘ฅ+1๏Šจ
  • E9๐‘ฅโˆ’3๐‘ฅโˆ’1๏Šจ

Q16:

Factorize fully (๐‘ฅโˆ’6๐‘ฆ)โˆ’216๐‘ฆ๏Šฉ๏Šฉ.

  • A(๐‘ฅโˆ’6๐‘ฆ)๏€น๐‘ฅ+6๐‘ฅ๐‘ฆ+36๐‘ฆ๏…๏Šจ๏Šจ
  • B(๐‘ฅโˆ’12๐‘ฆ)๏€น๐‘ฅ+6๐‘ฅ๐‘ฆ+36๐‘ฆ๏…๏Šจ๏Šจ
  • C(๐‘ฅโˆ’12๐‘ฆ)๏€น๐‘ฅโˆ’6๐‘ฅ๐‘ฆ+36๐‘ฆ๏…๏Šจ๏Šจ
  • D(๐‘ฅโˆ’6๐‘ฆ)๏€น๐‘ฅโˆ’6๐‘ฅ๐‘ฆ+36๐‘ฆ๏…๏Šจ๏Šจ
  • E(๐‘ฅโˆ’12๐‘ฆ)๏€น๐‘ฅ+6๐‘ฅ๐‘ฆโˆ’36๐‘ฆ๏…๏Šจ๏Šจ

Q17:

Factorize fully โˆ’192๐‘ฅ๐‘ฆโˆ’375๐‘ฆ๏Šฉ๏Šช.

  • Aโˆ’3๐‘ฆ(4๐‘ฅ+5๐‘ฆ)๏€น16๐‘ฅโˆ’20๐‘ฅ๐‘ฆโˆ’25๐‘ฆ๏…๏Šจ๏Šจ
  • Bโˆ’3๐‘ฆ(4๐‘ฅ+5๐‘ฆ)๏€น16๐‘ฅ+20๐‘ฅ๐‘ฆโˆ’25๐‘ฆ๏…๏Šจ๏Šจ
  • Cโˆ’3๐‘ฆ(4๐‘ฅ+5๐‘ฆ)๏€น16๐‘ฅ+20๐‘ฅ๐‘ฆ+25๐‘ฆ๏…๏Šจ๏Šจ
  • Dโˆ’3๐‘ฆ(4๐‘ฅ+5๐‘ฆ)๏€น16๐‘ฅโˆ’20๐‘ฅ๐‘ฆ+25๐‘ฆ๏…๏Šจ๏Šจ

Q18:

Factorize fully 729๐‘ฅโˆ’343๏Šฉ.

  • A๏€น81๐‘ฅ+49๏…(9๐‘ฅโˆ’7)๏Šจ
  • B(9๐‘ฅ+7)๏€น81๐‘ฅโˆ’63๐‘ฅ+49๏…๏Šจ
  • C(9๐‘ฅโˆ’7)๏€น81๐‘ฅ+63๐‘ฅ+49๏…๏Šจ
  • D(9๐‘ฅโˆ’7)๏€น81๐‘ฅโˆ’63๐‘ฅ+49๏…๏Šจ
  • E(9๐‘ฅ+7)(9๐‘ฅโˆ’7)๏Šจ

Q19:

If ๐‘ฆ+๐‘Ž=(๐‘ฆโˆ’3)๏€น๐‘ฆ+3๐‘ฆ+9๏…๏Šฉ๏Šจ, what is the value of ๐‘Ž?

Q20:

Factorize fully (9๐‘ฅ+7๐‘ฆ)โˆ’729๐‘ฅ๏Šฉ๏Šฉ.

  • A7๐‘ฆ๏€น81๐‘ฅโˆ’126๐‘ฅ๐‘ฆโˆ’49๐‘ฆ๏…๏Šจ๏Šจ
  • B7๐‘ฆ๏€น243๐‘ฅ+189๐‘ฅ๐‘ฆ+49๐‘ฆ๏…๏Šจ๏Šจ
  • C๐‘ฆ๏€น162๐‘ฅโˆ’126๐‘ฅ๐‘ฆ+49๐‘ฆ๏…๏Šจ๏Šจ
  • D7๐‘ฆ๏€น81๐‘ฅโˆ’126๐‘ฅ๐‘ฆ+49๐‘ฆ๏…๏Šจ๏Šจ
  • E7๐‘ฆ๏€น243๐‘ฅโˆ’189๐‘ฅ๐‘ฆ+49๐‘ฆ๏…๏Šจ๏Šจ

Q21:

Factorize fully (2๐‘ฅ+7๐‘ฆ)โˆ’(2๐‘ฅโˆ’7๐‘ฆ)๏Šฉ๏Šฉ.

  • A4๐‘ฅ๏€น4๐‘ฅ+147๐‘ฆ๏…๏Šจ๏Šจ
  • B14๐‘ฆ๏€น12๐‘ฅ+49๐‘ฆ๏…๏Šจ๏Šจ
  • C4๐‘ฅ๏€น12๐‘ฅ+49๐‘ฆ๏…๏Šจ๏Šจ
  • D14๐‘ฆ๏€น4๐‘ฅ+147๐‘ฆ๏…๏Šจ๏Šจ
  • Eโˆ’14๐‘ฆ๏€น12๐‘ฅ+49๐‘ฆ๏…๏Šจ๏Šจ

Q22:

If ๐‘ฅ+๐‘˜=(๐‘ฅ+๐‘˜)๏€น๐‘ฅโˆ’2๐‘ฅ+๐‘˜๏…๏Šฉ๏Šฉ๏Šจ๏Šจ, what is the value of ๐‘˜?

Q23:

Expand and simplify ๏€น๐‘ฅโˆ’5๏…๏€น๐‘ฅ+5๏…+52๏Šฉ๏Šฉ, and then factorize the result completely.

  • A๏€น๐‘ฅโˆ’3๏…๏€น๐‘ฅ+3๐‘ฅ+3๏…๏€น๐‘ฅโˆ’3๐‘ฅ+3๏…๏Šจ๏Šจ๏Šจ
  • B๏€น๐‘ฅโˆ’5๏…๏€น๐‘ฅ+5๐‘ฅ+25๏…๏Šจ๏Šช๏Šจ
  • C๏€น๐‘ฅ+3๏…๏€น๐‘ฅ+3๐‘ฅ+9๏…๏Šจ๏Šช๏Šจ
  • D๏€น๐‘ฅ+5๏…๏€น๐‘ฅโˆ’5๐‘ฅ+25๏…๏Šจ๏Šช๏Šจ
  • E๏€น๐‘ฅ+3๏…๏€น๐‘ฅ+3๐‘ฅ+3๏…๏€น๐‘ฅโˆ’3๐‘ฅ+3๏…๏Šจ๏Šจ๏Šจ

Q24:

Factorize fully 1216๐‘ฅโˆ’125๐‘ฆ๏Šฉ๏Šฉ.

  • A๏€ผ16๐‘ฅโˆ’100๐‘ฆ๏ˆ๏€ผ136๐‘ฅ+53๐‘ฅ๐‘ฆ+25๐‘ฆ๏ˆ๏Šจ๏Šจ
  • B๏€ผ16๐‘ฅโˆ’100๐‘ฆ๏ˆ๏€ผ136๐‘ฅโˆ’53๐‘ฅ๐‘ฆ+25๐‘ฆ๏ˆ๏Šจ๏Šจ
  • C๏€ผ16๐‘ฅโˆ’5๐‘ฆ๏ˆ๏€ผ136๐‘ฅ+56๐‘ฅ๐‘ฆ+25๐‘ฆ๏ˆ๏Šจ๏Šจ
  • D๏€ผ16๐‘ฅโˆ’5๐‘ฆ๏ˆ๏€ผ136๐‘ฅโˆ’56๐‘ฅ๐‘ฆ+25๐‘ฆ๏ˆ๏Šจ๏Šจ

Q25:

Factorize fully (๐‘ฅโˆ’๐‘ฆ)โˆ’216(๐‘ฅโˆ’๐‘ฆ)๏Šช.

  • A(๐‘ฅโˆ’๐‘ฆ)(๐‘ฅโˆ’๐‘ฆโˆ’6)๏€น(๐‘ฅโˆ’๐‘ฆ)+6(๐‘ฅโˆ’๐‘ฆ)โˆ’36๏…๏Šจ
  • B(๐‘ฅโˆ’๐‘ฆ)(๐‘ฅโˆ’๐‘ฆโˆ’6)๏€น(๐‘ฅโˆ’๐‘ฆ)+6(๐‘ฅโˆ’๐‘ฆ)+36๏…๏Šจ
  • C(๐‘ฅโˆ’๐‘ฆ)(๐‘ฅโˆ’๐‘ฆโˆ’6)๏€น(๐‘ฅโˆ’๐‘ฆ)โˆ’6(๐‘ฅโˆ’๐‘ฆ)โˆ’36๏…๏Šจ
  • D(๐‘ฅโˆ’๐‘ฆ)๏€น(๐‘ฅโˆ’๐‘ฆ)+6(๐‘ฅโˆ’๐‘ฆ)+36๏…๏Šจ
  • E(๐‘ฅโˆ’๐‘ฆ)(๐‘ฅโˆ’๐‘ฆโˆ’6)๏€น(๐‘ฅโˆ’๐‘ฆ)โˆ’6(๐‘ฅโˆ’๐‘ฆ)+36๏…๏Šจ

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