Question

Multiply or divide, as indicated. See Example 3. ((m² + 3m + 2) / (m² + 5m + 4)) ÷ ((m² + 5m + 6) / (m² + 10m + 24))

Accepted Answer

Rewrite the expression as a division of two rational expressions: $$\frac{m$$^{2}$$ + 3m + 2}{m$$^{2}$$ + 5m + 4} \div \frac{m$$^{2}$$ + 5m + 6}{m$$^{2}$$ + 10m + 24}. $$Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, rewrite the expression as $$\frac{m$$^{2}$$ + 3m + 2}{m$$^{2}$$ + 5m + 4} 	imes \frac{m$$^{2}$$ + 10m + 24}{m$$^{2}$$ + 5m + 6}. $$Factor each quadratic polynomial completely: 
- $$m^{2} + 3m + 2$ factors to (m + 1)(m + 2$$)$$,
- m$$^{2}$$ + 5m + 4$$ factors to $$(m + 1)(m + 4)$$,
- $$m^{2} + 5m + 6$ factors to (m + 2)(m + 3$$)$$,
- m$$^{2}$$ + 10m + 24$$ factors to $$(m + 4)(m + 6). $$Substitute the factored forms back into the expression: $$\frac{(m + 1)(m + 2)}{(m + 1)(m + 4)} 	imes \frac{(m + 4)(m + 6)}{(m + 2)(m + 3)}. $$Cancel out common factors in numerator and denominator across the multiplication, then multiply the remaining factors to write the simplified expression.

Multiply or divide, as indicated. See Example 3. ((m² + 3m + 2) / (m² + 5m + 4)) ÷ ((m² + 5m + 6) / (m² + 10m + 24))

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Key Concepts

Factoring Quadratic Expressions

Dividing Rational Expressions

Simplifying Rational Expressions