Simplifying Rational Expressions: (2x^2  x + 7) / (x + 5)
In algebra, simplifying rational expressions often involves factoring and canceling common factors. Let's explore how to simplify the expression (2x^2  x + 7) / (x + 5).
Factoring the Numerator
The numerator, 2x^2  x + 7, doesn't factor easily using traditional methods. This is because it doesn't have any real roots. Therefore, we can't simplify this expression further by factoring.
Polynomial Long Division
Since we can't factor the numerator, we can use polynomial long division to simplify the expression.
Steps for Polynomial Long Division:

Set up the division:
________ x+5  2x^2  x + 7

Divide the leading terms:
 The leading term of the divisor (x+5) is x.
 The leading term of the dividend (2x^2  x + 7) is 2x^2.
 2x^2 / x = 2x (Write this above the line)

Multiply the quotient by the divisor:
 2x (x + 5) = 2x^2 + 10x

Subtract the result from the dividend:
2x x+5  2x^2  x + 7 (2x^2 + 10x)  11x + 7

Bring down the next term:
2x x+5  2x^2  x + 7 (2x^2 + 10x)  11x + 7

Repeat steps 25 with the new dividend (11x + 7):
 11x / x = 11 (Write this above the line)
 11 (x + 5) = 11x  55
 Subtract:
2x  11 x+5  2x^2  x + 7 (2x^2 + 10x)  11x + 7 (11x  55)  62

The remainder is 62.
Simplified Expression
Therefore, the simplified expression is:
(2x^2  x + 7) / (x + 5) = 2x  11 + 62/(x + 5)
This means the original expression can be rewritten as a polynomial (2x  11) plus a fraction with the same denominator as the original expression, but with the remainder (62) as the numerator.
Important Note: The simplified expression is only valid for values of x where x ≠ 5. This is because the original expression is undefined when x = 5 since it would result in division by zero.