Factorise completely: 1. $$ t x-t y+r x-r y $$ 2. $$ b y-a x+b x-a y $$ 3. $$ 3 a^{2}-a x+6 a b-2 b x $$ 4. $$ x^{3}+3 x^{2}-x-3 $$ 5. $$ 2 a-1-2 a b+b $$ 6. $$ 6 x^{2}-a y-2 a x+3 x y $$ 7. $$ 4 a^{2}-4 b^{2}-b^{3}+a^{2} b $$ 8. $$ 6 x^{2}-x^{5}-6+x^{3} $$ 9. $$ x^{4}-x^{2}-x^{2} y^{2}+y^{2} $$ 10. $$ x^{3}+y^{3}+x+y $$

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UpStudy AI · Accepted Answer

Here are the complete factorizations for each of the given expressions:

1. **$$ t x - t y + r x - r y $$**
   
   Factor by grouping:
   $$
   t(x - y) + r(x - y) = (t + r)(x - y)
   $$
   
   **Factorized Form:** $$(t + r)(x - y)$$

2. **$$ b y - a x + b x - a y $$**
   
   Rearrange and factor:
   $$
   b(x + y) - a(x + y) = (b - a)(x + y)
   $$
   
   **Factorized Form:** $$(b - a)(x + y)$$

3. **$$ 3 a^{2} - a x + 6 a b - 2 b x $$**
   
   Factor by grouping:
   $$
   3a(a + 2b) - x(a + 2b) = (a + 2b)(3a - x)
   $$
   
   **Factorized Form:** $$(3a - x)(a + 2b)$$

4. **$$ x^{3} + 3x^{2} - x - 3 $$**
   
   Factor by grouping:
   $$
   x^{2}(x + 3) - 1(x + 3) = (x + 3)(x^{2} - 1) = (x + 3)(x + 1)(x - 1)
   $$
   
   **Factorized Form:** $$(x + 3)(x + 1)(x - 1)$$

5. **$$ 2 a - 1 - 2 a b + b $$**
   
   Rearrange and factor:
   $$
   (2a - 2ab) + (b - 1) = 2a(1 - b) + 1(b - 1) = (2a - 1)(1 - b)
   $$
   
   **Factorized Form:** $$(2a - 1)(1 - b)$$

6. **$$ 6 x^{2} - a y - 2 a x + 3 x y $$**
   
   Factor by grouping:
   $$
   2x(3x - a) + y(3x - a) = (3x - a)(2x + y)
   $$
   
   **Factorized Form:** $$(3x - a)(2x + y)$$

7. **$$ 4 a^{2} - 4 b^{2} - b^{3} + a^{2} b $$**
   
   Factor by grouping:
   $$
   (4a^{2} + a^{2}b) - (4b^{2} + b^{3}) = a^{2}(4 + b) - b^{2}(4 + b) = (4 + b)(a^{2} - b^{2}) = (4 + b)(a - b)(a + b)
   $$
   
   **Factorized Form:** $$(b + 4)(a - b)(a + b)$$

8. **$$ 6 x^{2} - x^{5} - 6 + x^{3} $$**
   
   Rearrange and factor:
   $$
   -x^{5} + x^{3} + 6x^{2} - 6 = -(x^5 - x^3 - 6x^2 + 6)
   $$
   
   Factor $$x^5 - x^3 - 6x^2 + 6$$ by finding roots:
   $$
   x^5 - x^3 - 6x^2 + 6 = (x - 1)(x + 1)(x^3 - 6)
   $$
   
   **Factorized Form:** $$- (x - 1)(x + 1)(x^3 - 6)$$

9. **$$ x^{4} - x^{2} - x^{2} y^{2} + y^{2} $$**
   
   Factor by grouping:
   $$
   x^{4} - x^{2} y^{2} - x^{2} + y^{2} = (x^{2} - 1)(x^{2} - y^{2}) = (x - 1)(x + 1)(x - y)(x + y)
   $$
   
   **Factorized Form:** $$(x - 1)(x + 1)(x - y)(x + y)$$

10. **$$ x^{3} + y^{3} + x + y $$**
    
    Factor by grouping:
    $$
    x^{3} + y^{3} + x + y = (x + y)(x^{2} - xy + y^{2}) + (x + y) = (x + y)(x^{2} - xy + y^{2} + 1)
    $$
    
    **Factorized Form:** $$(x + y)(x^{2} - xy + y^{2} + 1)$$
1. $$ t x - t y + r x - r y = (t + r)(x - y) $$ 2. $$ b y - a x + b x - a y = (b - a)(x + y) $$ 3. $$ 3 a^{2} - a x + 6 a b - 2 b x = (3a - x)(a + 2b) $$ 4. $$ x^{3} + 3x^{2} - x - 3 = (x + 3)(x + 1)(x - 1) $$ 5. $$ 2 a - 1 - 2 a b + b = (2a - 1)(1 - b) $$ 6. $$ 6 x^{2} - a y - 2 a x + 3 x y = (3x - a)(2x + y) $$ 7. $$ 4 a^{2} - 4 b^{2} - b^{3} + a^{2} b = (b + 4)(a - b)(a + b) $$ 8. $$ 6 x^{2} - x^{5} - 6 + x^{3} = - (x - 1)(x + 1)(x^3 - 6) $$ 9. $$ x^{4} - x^{2} - x^{2} y^{2} + y^{2} = (x - 1)(x + 1)(x - y)(x + y) $$ 10. $$ x^{3} + y^{3} + x + y = (x + y)(x^{2} - xy + y^{2} + 1) $$

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