Do not divide out factors of individual terms
Avoid illegal canceling in Grade 10 algebra: learn to factor the entire numerator and denominator first before canceling common factors in rational expressions.
Key Concepts
You cannot simplify by canceling parts of terms that are being added or subtracted. For example, $$\frac{4ax + 3x}{2ax x} \neq \frac{4+3x}{2 x}$$.
Wrong: $$\frac{6x+12}{6} \neq x+12$$. Correct: $$\frac{6(x+2)}{6} = x+2$$ Wrong: $$\frac{x^2 16}{x 4} \neq x 4$$. Correct: $$\frac{(x 4)(x+4)}{x 4} = x+4$$.
Hold on! You can't just slash parts of terms connected by addition or subtraction. It’s like trying to cancel the 'utter' out of 'butterfly' and 'mutter' – it makes no sense! To simplify correctly, you must first factor the entire numerator and denominator. Only then can you cancel out common factors that are being multiplied.
Common Questions
Why can't you cancel terms connected by addition or subtraction?
Addition and subtraction 'glue' terms together. You can only cancel factors connected by multiplication. For example, (6x+12)/6 ≠ x+12; you must factor first to get 6(x+2)/6 = x+2.
What is the correct process to simplify a rational expression?
Factor the numerator completely, factor the denominator completely, then cancel identical factors that appear on both top and bottom. Never cancel individual terms.
How do you simplify (x²-25)/(x+5)?
Factor the numerator: x²-25=(x-5)(x+5). Then cancel the common factor (x+5) from numerator and denominator, leaving x-5 as the simplified result.