Multiply and Simplify $$\frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 17 x + 60 } \cdot \frac { x ^ { 2 } - 13 x + 40 } { x ^ { 2 } - 9 x + 14 }$$

question 187

Multiple Choice

Multiply and simplify.
- $$\frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 17 x + 60 } \cdot \frac { x ^ { 2 } - 13 x + 40 } { x ^ { 2 } - 9 x + 14 }$$

• A) \(\frac { ( x - 3 ) ( x - 8 ) } { ( x - 12 ) ( x - 7 ) }\)
• B) \(\frac { \left( x ^ { 2 } - 5 x + 6 \right) \left( x ^ { 2 } - 13 x + 40 \right) } { \left( x ^ { 2 } - 17 x + 60 \right) \left( x ^ { 2 } - 9 x + 14 \right) }\)
• C) \(\frac { ( x + 3 ) ( x + 8 ) } { ( x + 12 ) ( x + 7 ) }\)
• D) \(\frac { ( x - 3 ) } { ( x - 7 ) }\)

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Definitions:

Existential Generalization

A logical inference rule stating that if something is true for some member of a domain, then it is true that there exists something in that domain for which the proposition holds.

Quantifier Negation

The process of changing the quantifier in a statement, such as from "all" to "not all," to negate its original meaning.

Universal Generalization

A principle of deductive reasoning that allows one to conclude that something is true for all cases after confirming it is true in every observed case.

Existential Instantiation

A logical principle that states from the knowledge that something exists, one can infer the existence of an instance of that thing.