Simplifying Rational Expressions
Simplify rational expressions in Grade 9 algebra by factoring both numerator and denominator, canceling common polynomial factors, and stating domain restrictions where the denominator equals zero.
Key Concepts
Property To simplify a rational expression, factor the numerator and denominator. Then, divide out any common factors that appear in both.
Examples $\frac{10x^3}{15x} = \frac{5x(2x^2)}{5x(3)} = \frac{2x^2}{3}$ $\frac{3y 9}{y 3} = \frac{3(y 3)}{(y 3)} = 3$ $\frac{5z^2+10z}{z+2} = \frac{5z(z+2)}{(z+2)} = 5z$.
Explanation This is like a mathematical magic trick where you make things disappear! By factoring, you reveal the secret identical parts of the top and bottom. Once you find a matching pair, you can cancel them out, leaving you with a much simpler, tidier expression. Presto, chango, simplified!
Common Questions
What is the process for simplifying a rational expression step by step?
Factor the numerator completely. Factor the denominator completely. Identify factors that appear in both (common factors). Cancel those common factors. State any excluded values from the original denominator.
How do you simplify (x² + 5x + 6) / (x² + 2x - 3)?
Factor numerator: (x+2)(x+3). Factor denominator: (x+3)(x-1). Cancel the common factor (x+3): result is (x+2)/(x-1), where x ≠ -3 and x ≠ 1.
Why is it incorrect to cancel individual terms rather than factors?
Terms separated by addition or subtraction cannot be cancelled — only complete multiplicative factors can. Cancelling x from (x+2)/x is wrong; only factor (x+2)/(x·something) would allow cancellation of x.