Simplifying Rational Expressions
To simplify a rational expression, factor both numerator and denominator completely, then cancel common factors.
Simplification Steps
- Factor numerator completely
- Factor denominator completely
- Identify and cancel common factors
- State any excluded values (values that make denominator zero)
Problem (i): Simplify \(\frac{x^2 - 1}{x^2 + x}\)
1
Factor numerator:
x² - 1 = (x + 1)(x - 1) ℹ️
2
Factor denominator:
x² + x = x(x + 1)
3
Rewrite expression:
\(\frac{(x + 1)(x - 1)}{x(x + 1)}\)
4
Cancel common factor (x + 1):
\(\frac{x - 1}{x}\)
Simplified form: \(\frac{x - 1}{x}\)
Excluded values: x ≠ 0, -1 (would make denominator zero)
Problem (ii): Simplify \(\frac{x^2 - 11x + 18}{x^2 - 4x + 4}\)
1
Factor numerator:
x² - 11x + 18 = (x - 2)(x - 9)
2
Factor denominator:
x² - 4x + 4 = (x - 2)² ℹ️
3
Rewrite expression:
\(\frac{(x - 2)(x - 9)}{(x - 2)^2}\)
4
Cancel common factor (x - 2):
\(\frac{x - 9}{x - 2}\)
Simplified form: \(\frac{x - 9}{x - 2}\)
Excluded values: x ≠ 2 (would make denominator zero)
Finding Excluded Values
Excluded values make the denominator zero (which is undefined in mathematics).
How to Find Excluded Values
- Set denominator equal to zero
- Solve the resulting equation
- These solutions are the excluded values
- Always check simplified form's denominator too!
Problem (i): Find excluded values for \(\frac{y}{y^2 - 25}\)
1
Set denominator equal to zero:
y² - 25 = 0
2
Solve for y:
y² = 25
y = ±5
Excluded values: y ≠ 5, -5
Problem (iii): Find excluded values for \(\frac{x^2 + 6x + 8}{x^2 + x - 2}\)
1
Factor denominator:
x² + x - 2 = (x + 2)(x - 1)
2
Set denominator equal to zero:
(x + 2)(x - 1) = 0
3
Solve for x:
x + 2 = 0 → x = -2
x - 1 = 0 → x = 1
Excluded values: x ≠ -2, 1
4
Bonus: Check simplified form:
Numerator: x² + 6x + 8 = (x + 2)(x + 4)
Simplified form: \(\frac{x + 4}{x - 1}\) (still excludes x = 1)