Simplifying by Dividing Out Common Factors
Simplify rational expressions in Grade 10 algebra by fully factoring numerator and denominator, then dividing out common factors to reach the lowest terms expression.
Key Concepts
A rational expression is simplified when there are no common factors, other than 1, in the numerator and denominator. This is achieved by dividing out common factors.
Excluded value is 3. Simplify: $$\frac{3x + 9}{6x + 18} = \frac{3(x + 3)}{6(x + 3)} = \frac{3}{6} = \frac{1}{2}$$Excluded value is 2. Simplify: $$\frac{5c^2 20}{c + 2} = \frac{5(c^2 4)}{c + 2} = \frac{5(c+2)(c 2)}{c+2} = 5(c 2)$$.
This is like a matching game for fractions! First, find any 'excluded values' that make the denominator zero. Then, factor the top and bottom expressions completely. If you spot the exact same factor on both levels, you can cancel them out. It’s the ultimate clean up move for messy polynomial fractions, revealing a much simpler expression underneath.
Common Questions
What is the process to simplify a rational expression?
Factor the numerator completely, factor the denominator completely, then cancel any factors that appear identically in both numerator and denominator.
How do you simplify (x²-9)/(x+3)?
Factor numerator: x²-9 = (x-3)(x+3). Cancel the common factor (x+3): (x-3)(x+3)/(x+3) = x-3, with the restriction x ≠ -3.
Why must you state excluded values after simplifying?
The simplified expression has a different domain than the original. Values that made the original denominator zero must be excluded, even though the factor no longer appears after simplification.