# 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 - 3x∙ $$\frac{1}{x}$$ ∙ $$\frac{1}{x^{2}}$$ - $$\frac{1}{x^{3}}$$ - 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)- 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.

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