Simplifying by Factoring Out -1
Simplify rational expressions with opposite binomials in Grade 10 by factoring out -1 to flip signs, match factors, and cancel terms like (x-5) and (5-x).
Key Concepts
To simplify expressions with opposite binomials, such as $\frac{a b}{b a}$, factor out a $ 1$ from one of the binomials. For example, $(b a)$ can be rewritten as $ 1(a b)$.
Excluded value is 4. Simplify: $$\frac{5x 20}{8 2x} = \frac{5(x 4)}{2(4 x)} = \frac{5(x 4)}{ 2(x 4)} = \frac{5}{2}$$Excluded value is 3. Simplify: $$\frac{y^2 9}{3 y} = \frac{(y 3)(y+3)}{ (y 3)} = (y+3)$$.
Ever see factors that are almost identical but backward, like $(x 5)$ and $(5 x)$? Don't worry! Use a clever trick by factoring out a $ 1$ from one of them. This flips its signs, making it a perfect match for the other factor. Now you can cancel them out, leaving just a $ 1$ behind. It's a ninja move for simplifying!
Common Questions
Why do we factor out -1 when simplifying opposite binomials?
Opposite binomials like (x-5) and (5-x) are not identical and cannot be directly canceled. Factoring -1 from one, e.g. (5-x) = -1(x-5), makes them match so they can cancel.
How do you simplify (x²-25)/(5-x)?
Factor the numerator: (x-5)(x+5). Rewrite denominator as -1(x-5). Cancel (x-5) to get (x+5)/(-1) = -(x+5).
What is a critical error students make with opposite binomials?
Canceling (x-5) and (5-x) without accounting for the -1 factor. This flips the sign of the answer. Always track the -1 that remains after canceling.