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

9th Grade Math

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