Simplifying Rational Expressions with Trinomials
Simplify rational expressions containing trinomials in Grade 10 algebra by factoring numerator and denominator completely, then canceling common binomial factors.
Key Concepts
To simplify rational expressions involving trinomials, first factor the trinomials in the numerator and denominator into binomials. Then, identify and divide out any common binomial factors. Always determine the excluded values from the original denominator before simplifying.
Excluded values are 2 and 2. Simplify: $$\frac{2x^2 + 2x 12}{3x^2 12} = \frac{2(x^2+x 6)}{3(x^2 4)} = \frac{2(x+3)(x 2)}{3(x+2)(x 2)} = \frac{2(x+3)}{3(x+2)}$$Excluded values are 1 and 3. Simplify: $$\frac{x^2 5x + 6}{x^2 4x + 3} = \frac{(x 2)(x 3)}{(x 1)(x 3)} = \frac{x 2}{x 1}$$.
When you see trinomials on the top and bottom of a fraction, it’s time to put your factoring skills to the test! Break down each trinomial into its binomial factors, just like solving a puzzle. Once everything is factored, you can easily spot and cancel out any matching binomial pairs, turning a complicated expression into something much cleaner.
Common Questions
How do you simplify a rational expression with trinomials in numerator and denominator?
Factor both the numerator and denominator trinomials completely, then identify and cancel any common binomial factors between them.
How do you factor the trinomial x² + 5x + 6?
Find two numbers that multiply to 6 and add to 5: those are 2 and 3. So x² + 5x + 6 = (x+2)(x+3).
What are excluded values in a simplified rational expression?
Excluded values are x-values that make the original denominator equal zero, even if those factors canceled. They must be stated as restrictions in the simplified form.