We are asked to decide whether the expression x^2-9x+3 is the same as x-3. To do so, let's examine these expressions one by one.
First Expression
Let's start with x^2-9x+3.
Expression
Is It a Polynomial?
Numerator
x^2-9
Yes
Denominator
x+3
Yes
Both the numerator and the denominator of the given expression are polynomials, so x^2-9x+3 is a rational expression. Recall that rational expressions are undefined for the values of variable which make the denominator equal 0. These values are called excluded values. Let's find the excluded values of our expression.
x+3=0 ⇔ x=- 3
We have that the expression x^2-9x+3 is defined for all real numbersexcept x=- 3. Now we will write our rational expression in simplified form.
The simplified form of x^2-9x+3 is x-3. Keep in mind that the obtained expression is undefined for x=- 3. Let's put in a table the information gathered so far.
Given Expression
Simplified Form
Excluded Values
x^2-9/x+3
x-3
x=- 3
x-3
Second Expression
The second expression that we are given, x-3, cannot be simplified any further. Additionally, this expression is a polynomial, so it is defined for all real numbers. In other words, x-3 has no excluded values. Finally, we will complete our table.
Given Expression
Simplified Form
Excluded Values
x^2-9/x+3
x-3
x=- 3
x-3
x-3
None
Conclusion
Even though x^2-9x+3 simplifies to x-3, writing x-3 is not the same as writing x^2-9x+3. This is because x^2-9x+3 is undefined when x=- 3 meanwhile x-3 is defined when x=- 3.
