Question: examples of algebraic fractions multiplication and division
make it longer and harder question
1. Let's start with a more complex example of multiplication of algebraic fractions.
Given: $$\frac{3x^2 - 6x}{4x} \times \frac{8x^3}{12x^2 - 18x}$$
2. The formula for multiplying fractions is:
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
3. First, factor all polynomials where possible:
$$3x^2 - 6x = 3x(x - 2)$$
$$12x^2 - 18x = 6x(2x - 3)$$
4. Rewrite the multiplication with factored forms:
$$\frac{3x(x - 2)}{4x} \times \frac{8x^3}{6x(2x - 3)}$$
5. Multiply numerators and denominators:
$$\frac{3x(x - 2) \times 8x^3}{4x \times 6x(2x - 3)} = \frac{24x^4(x - 2)}{24x^2(2x - 3)}$$
6. Simplify the fraction by canceling common factors:
$$\frac{\cancel{24}x^{4 - 2}(x - 2)}{\cancel{24}(2x - 3)} = \frac{x^2(x - 2)}{2x - 3}$$
7. So, the product simplifies to:
$$\frac{x^2(x - 2)}{2x - 3}$$
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8. Now, let's do a more challenging example of division of algebraic fractions.
Given: $$\frac{5x^3 - 10x^2}{15x} \div \frac{10x^2 - 20x}{25x^3}$$
9. The formula for dividing fractions is:
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$
10. Factor all polynomials:
$$5x^3 - 10x^2 = 5x^2(x - 2)$$
$$10x^2 - 20x = 10x(x - 2)$$
11. Rewrite the division as multiplication:
$$\frac{5x^2(x - 2)}{15x} \times \frac{25x^3}{10x(x - 2)}$$
12. Multiply numerators and denominators:
$$\frac{5x^2(x - 2) \times 25x^3}{15x \times 10x(x - 2)} = \frac{125x^5(x - 2)}{150x^2(x - 2)}$$
13. Simplify by canceling common factors including $x - 2$ and numeric factors:
$$\frac{\cancel{125}x^{5 - 2}\cancel{(x - 2)}}{\cancel{150}x^{2}\cancel{(x - 2)}} = \frac{\frac{125}{25}x^{3}}{\frac{150}{25}} = \frac{5x^{3}}{6}$$
14. So, the quotient simplifies to:
$$\frac{5x^{3}}{6}$$