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)
Problem (iii): Simplify \(\frac{9x^2 + 81x}{x^3 + 8x^2 - 9x}\)
1
Factor numerator:
9x² + 81x = 9x(x + 9)
2
Factor denominator:
x³ + 8x² - 9x = x(x² + 8x - 9) = x(x + 9)(x - 1)
3
Rewrite expression:
\(\frac{9x(x + 9)}{x(x + 9)(x - 1)}\)
4
Cancel common factors (x and x+9):
\(\frac{9}{x - 1}\)
Simplified form: \(\frac{9}{x - 1}\)
Excluded values: x ≠ 0, -9, 1 (would make original denominator zero)
Problem (iv): Simplify \(\frac{p^2 - 3p - 40}{2p^3 - 24p^2 + 64p}\)
1
Factor numerator:
p² - 3p - 40 = (p - 8)(p + 5)
2
Factor denominator:
2p³ - 24p² + 64p = 2p(p² - 12p + 32) = 2p(p - 8)(p - 4)
3
Rewrite expression:
\(\frac{(p - 8)(p + 5)}{2p(p - 8)(p - 4)}\)
4
Cancel common factor (p - 8):
\(\frac{p + 5}{2p(p - 4)}\)
Simplified form: \(\frac{p + 5}{2p(p - 4)}\)
Excluded values: p ≠ 0, 4, 8 (would make original 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 (ii): Find excluded values for \(\frac{t}{t^2 - 5t + 6}\)
1
Factor denominator:
t² - 5t + 6 = (t - 2)(t - 3)
2
Set denominator equal to zero:
(t - 2)(t - 3) = 0
3
Solve for t:
t - 2 = 0 → t = 2
t - 3 = 0 → t = 3
Excluded values: t ≠ 2, 3
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)
Problem (iv): Find excluded values for \(\frac{x^3 - 27}{x^3 + x^2 - 6x}\)
1
Factor denominator:
x³ + x² - 6x = x(x² + x - 6) = x(x + 3)(x - 2)
2
Set denominator equal to zero:
x(x + 3)(x - 2) = 0
3
Solve for x:
x = 0, -3, 2
Excluded values: x ≠ -3, 0, 2
4
Note about numerator:
x³ - 27 = (x - 3)(x² + 3x + 9) has no common factors with denominator