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Worksheet on Factorization
Practice the questions given in the Worksheet on Factorization.
Problems on Factorization of expressions of the form a\(^{3}\) ± b\(^{3}\)
1. Factorize:
(i) 8x\(^{3}\) + 27y\(^{3}\)
(ii) 216a\(^{3}\) + 1
(iii) a\(^{6}\) + 1
(iv) x\(^{3}\) + \(\frac{1}{x^{3}}\)
(v) a\(^{3}\) + 8b\(^{6}\)
2. Factorize:
(i) 1 – 729m\(^{3}\)
(ii) 125x\(^{3}\) – 27y\(^{3}\)
(iii) a\(^{3}\) - \(\frac{8}{b^{3}}\)
(iv) x\(^{6}\) – y\(^{3}\)
3. Factorize:
(i) x\(^{6}\) - 1
(ii) a\(^{6}\) – 729b\(^{6}\)
4. Factorize a\(^{6}\) + b\(^{6}\) and prove that its value is zero if a\(^{4}\) + b\(^{4}\) = a\(^{2}\)b\(^{2}\).
On Factorization of expressions reducible to a\(^{3}\) ± b\(^{3}\)from
5. Factorize:
(i) x\(^{3}\) + 3x\(^{2}\) + 3x + 28
(ii) a\(^{3}\) + 3a\(^{2}\) + 3a - 7
(iii) x\(^{3}\) – 3x – 1 + \(\frac{3}{x}\) - \(\frac{1}{x^{3}}\)
[Hint: Given expression = x3 - 3x2 ∙ \(\frac{1}{x}\) ∙ \(\frac{1}{x^{2}}\) - \(\frac{1}{x^{3}}\) - 1 = (x - \(\frac{1}{x}\))3 - 13.]
(iv) a\(^{3}\) + 7b\(^{3}\) + 6ab(a + 2b)
[Hint: Given expression = a3 + (2b)3 + 3 ∙ a ∙ 2b(a + 2b) - b3 = (a + 2b)3 - b3.]
Factorization of expressions of the form a\(^{3}\) + b\(^{3}\) + c\(^{3}\) – 3abc
6. Factorize:
(i) 8 + x\(^{3}\) + y\(^{3}\) – 6xy
(ii) a\(^{3}\) + 8b\(^{3}\) + 27c\(^{3}\) – 18abc
Problems on Miscellaneous Factorization
7. Factorize:
(i) (1 – x)\(^{3}\) + (y – 1)\(^{3}\) + (x – y)\(^{3}\)
(ii) (2a – b – c)\(^{3}\) + (2b – c – a)\(^{3}\) + (2c – a – b)\(^{3}\)
8. Factorize:
(i) x\(^{9}\) + 1
(ii) a\(^{12}\) – b\(^{12}\)
(iii) (a + b)\(^{3}\) + 8(a – b)\(^{3}\)
(iv) a\(^{9}\) – b\(^{9}\)
9. Factorize:
(i) x\(^{3}\) + x\(^{2}\) - 2
[Hint: Given expression = x3 - 1 + x2 - 1 (x - 1)(x2 + x + 1) + (x - 1)(x + 1) = (x - 1)(x2 + x + 1 + x + 1) = (x - 1)(x2 + 2x + 2).]
(ii) a\(^{3}\) + a\(^{2}\) - \(\frac{1}{a^{2}}\) - \(\frac{1}{a^{3}}\)
[Hint: Given expression = a3 - \(\frac{1}{a^{3}}\) + a2 \(\frac{1}{a^{2}}\) = (a - \(\frac{1}{a}\))(a2 + 1 + \(\frac{1}{a^{2}}\)) + (a - \(\frac{1}{a}\))(a + \(\frac{1}{a}\))].
Application problems on Factorization
10. (i) If a + \(\frac{1}{a}\) = 2, find a\(^{3}\) + \(\frac{1}{a^{3}}\).
(ii) If x - \(\frac{1}{x}\) = √3, find x\(^{3}\) - \(\frac{1}{x^{3}}\).
(iii) If m + \(\frac{1}{m}\) = √3, find m\(^{6}\) - \(\frac{1}{m^{6}}\).
[Hint: Given expression = m3 + \(\frac{1}{m^{3}}\) = (m + \(\frac{1}{m}\))3 - 3m ∙ \(\frac{1}{m}\) ∙ (m + \(\frac{1}{m}\)) = (√3)3 - 3√3 = 0.
And m6 + \(\frac{1}{m^{6}}\) = (m3 + \(\frac{1}{m^{3}}\))(m3 - \(\frac{1}{m^{3}}\)) = 0.]
11. (i) If x + y + z = 6, xyz = 6 and xy + yz + zx = 11 then find x\(^{3}\) + y\(^{3}\) + z\(^{3}\).
[Hint: Use x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - yz - zx - xy) = (x + y + z){(x + y + z)2 - 3(yz + zx + xy)}.]
(ii) If l + m + n = 9, l\(^{2}\)+ m\(^{2}\) + n\(^{2}\) = 27 and l\(^{3}\) + m\(^{3}\) + n\(^{3}\) = 81 then find lmn.
[Hint: Use l3 + m3 + n3 - 3lmn = (l + m + n)(l2 + m2 + n2 - mn - nl - lm)
and (l + y + z)2 - (l2 + m2 + n2) = 2(mn + nl + lm)}.]
12. Evaluate:
(i) \(\frac{361^{3} + 139^{3}}{361^{2} – 361 × 139 + 139^{2}}\)
(ii) \(\frac{272^{3} - 122^{3}}{136^{2} + 136 × 61 + 61^{2}}\)
13. Find the LCM and HCF.
(i) p\(^{3}\) + 8 and p\(^{2}\) + 4
(ii) 1 – 8x\(^{3}\), 1 – 4x\(^{2}\) and 1 – x – 2x\(^{2}\)
Answers for the Worksheet on Factorization are given below.
Answers:
1. (i) (2x + 3y)(4x\(^{2}\) – 6xy + 9y\(^{2}\))
(ii) (6a + 1)(36a\(^{2}\) – 6a + 1)
(iii) (a\(^{2}\) + 1)(a\(^{4}\) – a\(^{2}\) + 1)
(iv) (x + \(\frac{1}{x}\))(x\(^{2}\) – 1 + \(\frac{1}{x^{2}}\)
(v) (a + 2b)\(^{2}\)(a\(^{2}\) – 2ab\(^{2}\) + 4b\(^{4}\))
2. (i) (1 + 9m)(1 + 9m + 81m\(^{2}\))
(ii) (5x – 3y)(25x\(^{2}\) + 15xy + 9y\(^{2}\))
(iii) (a - \(\frac{2}{b}\))(a\(^{2}\) + \(\frac{2a}{b}\) + \(\frac{4}{b^{2}}\)
(iv) (x\(^{2}\) – y)(x\(^{4}\) + x\(^{2}\)y + y\(^{2}\))
3. (i) (x + 1)(x – 1)(x\(^{2}\) + x + 1)(x\(^{2}\) – x + 1)
(ii) (a + 3b)(a – 3b)(a\(^{2}\) + 3ab + 9b\(^{2}\))(a\(^{2}\) – 3ab + 9b\(^{2}\))
4. (a\(^{2}\) + b\(^{2}\))(a\(^{4}\) – a\(^{2}\)b\(^{2}\) + b\(^{4}\))
5. (i) (x + 4)(x\(^{2}\) – x + 7)
(ii) (a – 1)(a\(^{2}\) + 4a + 7)
(iii) (x - \(\frac{1}{x}\) – 1)(x\(^{2}\) + x - \(\frac{1}{x}\) + \(\frac{1}{x^{2}}\) – 1)
(iv) (a + b)(a\(^{2}\) + 5ab + 7b\(^{2}\))
6. (i) (2 + x + y)(4 + x\(^{2}\) + y\(^{2}\) – 2x – 2y – xy)
(ii) (a + 2b + 3c)(a\(^{2}\) + 4b\(^{2}\) + 9c\(^{2}\) – 2ab – 3ca – 6bc)
7. (i) 3(1 – x)(y – 1)(x – y)
(ii) 3(2a – b – c)(2b – c – a)(2c – a – b)
8. (i) (x + 1)(x\(^{2}\) – x + 1)(x\(^{6}\) – x\(^{3}\) + 1)
(ii) (a + b)(a – b)(a\(^{2}\) + b\(^{2}\))(a\(^{2}\) – ab + b\(^{2}\))(a\(^{2}\) + ab + b\(^{2}\))(a\(^{4}\) –a\(^{2}\)b\(^{2}\) + b\(^{4}\))
(iii) (3a – b)(3a\(^{2}\) – 10ab + 7b\(^{2}\))
(iv) (a – b)(a\(^{2}\) + ab + b\(^{2}\))(a\(^{6}\) + a\(^{3}\)b\(^{3}\) + b\(^{6}\))
9. (i) (x – 1)(x\(^{2}\) + 2x + 2)
(ii) (a - \(\frac{1}{a}\))(a\(^{2}\) + a + 1 + \(\frac{1}{a}\) + \(\frac{1}{a^{2}}\))
10. (i) 2
(ii) 6√3
(iii) 0
11. (i) 36
(ii) 27
12. (i) 500
(ii) 600
13. (i) LCM = (p + 2)(p – 2)(p\(^{2}\) – 2p + 4); HCF = p + 2
(ii) LCM = (1 + x)(1 – 2x)(1 + 2x)(1 + 2x + 4x\(^{2}\)); HCF = 1 – 2x
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