Factorise the following expressions :
`(i) ax-ay+bx-by " " (ii) x^(2)-x-ax+a " " (iii) x^(4)+x^(3)+x^(2)+x`
`(iv) 16(a+b)^(2)-4a-4b " " (v) x^(2)+(1)/(x^(2))+2-3x-(3)/(x) " " (vi) x^(2)-((a)/(b)+(b)/(a))x+1`
`(vii) x^(2)+(a-(1)/(a))x-1 " " (viii)ab(x^(2)+y^(2)+xy(a^(2)+b^(2)) " "(ix) (ax+by)^(2)+(bx-ay)^(2)`

Answer

Let's factorize the given expressions step by step.

### (i) Factorize \( ax - ay + bx - by \)

1. **Group the terms**:
\[ (ax - ay) + (bx - by) \]
2. **Factor out common terms**:
\[ a(x - y) + b(x - y) \]
3. **Factor out \((x - y)\)**:
\[ (x - y)(a + b) \]

**Final Answer**: \((x - y)(a + b)\)

### (ii) Factorize \( x^2 - x - ax + a \)

1. **Group the terms**:
\[ (x^2 - x) + (-ax + a) \]
2. **Factor out common terms**:
\[ x(x - 1) - a(x - 1) \]
3. **Factor out \((x - 1)\)**:
\[ (x - 1)(x - a) \]

**Final Answer**: \((x - 1)(x - a)\)

### (iii) Factorize \( x^4 + x^3 + x^2 + x \)

1. **Factor out \(x\)**:
\[ x(x^3 + x^2 + x + 1) \]
2. **Group the polynomial**:
\[ x(x^3 + 1 + x^2 + x) \]
3. **Factor \(x^3 + 1\)** using the sum of cubes:
\[ x(x + 1)(x^2 - x + 1) \]

**Final Answer**: \(x(x + 1)(x^2 - x + 1)\)

### (iv) Factorize \( 16(a + b)^2 - 4a - 4b \)

1. **Rewrite the expression**:
\[ 4[4(a + b)^2 - a - b] \]
2. **Group the terms**:
\[ 4[4(a + b)^2 - (a + b)] \]
3. **Factor out \((a + b)\)**:
\[ 4(a + b)(4(a + b) - 1) \]

**Final Answer**: \(4(a + b)(4(a + b) - 1)\)

### (v) Factorize \( x^2 + \frac{1}{x^2} + 2 - 3x - \frac{3}{x} \)

1. **Rewrite the expression**:
\[ (x^2 - 3x + 2) + \left(\frac{1}{x^2} - \frac{3}{x} + 1\right) \]
2. **Factor the first part**:
\[ (x - 1)(x - 2) \]
3. **Factor the second part**:
\[ \left(\frac{1}{x} - 1\right)^2 \]
4. **Combine**:
\[ (x - 1)(x - 2) + \left(\frac{1 - x}{x}\right)^2 \]

**Final Answer**: \((x - 1)(x - 2) + \left(\frac{1 - x}{x}\right)^2\)

### (vi) Factorize \( x^2 - \left(\frac{a}{b} + \frac{b}{a}\right)x + 1 \)

1. **Rewrite the expression**:
\[ x^2 - \left(\frac{a^2 + b^2}{ab}\right)x + 1 \]
2. **Use the quadratic formula to find roots**:
\[ x = \frac{\frac{a^2 + b^2}{ab} \pm \sqrt{\left(\frac{a^2 + b^2}{ab}\right)^2 - 4}}{2} \]
3. **Factor using the roots**:
\[ \left(x - \frac{a}{b}\right)\left(x - \frac{b}{a}\right) \]

**Final Answer**: \(\left(x - \frac{a}{b}\right)\left(x - \frac{b}{a}\right)\)

### (vii) Factorize \( x^2 + \left(a - \frac{1}{a}\right)x - 1 \)

1. **Use the quadratic formula**:
\[ x = \frac{-(a - \frac{1}{a}) \pm \sqrt{(a - \frac{1}{a})^2 + 4}}{2} \]
2. **Factor using roots**:
\[ \left(x - r_1\right)\left(x - r_2\right) \]

**Final Answer**: \(\left(x - r_1\right)\left(x - r_2\right)\)

### (viii) Factorize \( ab(x^2 + y^2 + xy(a^2 + b^2)) \)

1. **Rewrite the expression**:
\[ ab\left(x^2 + y^2 + xy(a^2 + b^2)\right) \]
2. **Group terms**:
\[ ab\left((x + ay)(x + by)\right) \]

**Final Answer**: \(ab\left((x + ay)(x + by)\right)\)

### (ix) Factorize \( (ax + by)^2 + (bx - ay)^2 \)

1. **Expand the squares**:
\[ a^2x^2 + 2abxy + b^2y^2 + b^2x^2 - 2abxy + a^2y^2 \]
2. **Combine like terms**:
\[ (a^2 + b^2)(x^2 + y^2) \]

**Final Answer**: \((a^2 + b^2)(x^2 + y^2)\)

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