4
Bringing fractions to a common denominator
Textbook: pp. 18–21
GoalFind the common denominator of algebraic fractions and bring fractions to it using additional factors.
New words
common denominator · umumiy maxrajadditional factor · qo‘shimcha ko‘paytuvchileast common multiple · eng kichik umumiy karralifactoring the denominators · maxrajlarni ko‘paytuvchilarga ajratish
Explanation
Before adding or subtracting fractions we bring them to a common denominator. For algebraic fractions we take the least common multiple of the denominators: the LCM of the numerical coefficients and the highest power of every letter or bracket factor. For this, factor the denominators first. Dividing the common denominator by the denominator of each fraction gives the additional factor for that fraction; multiply both its numerator and denominator by it. Watch the signs: b - a = -(a - b), so 1/(b - a) = -1/(a - b).
Worked examples
x/(6a²b) and y/(4ab³): common denominator 12a²b³. Additional factors 2b² and 3a: x/(6a²b) = 2xb²/(12a²b³), y/(4ab³) = 3ay/(12a²b³).
1/(x - 2) and 3/(x² - 4): x² - 4 = (x - 2)(x + 2), so the common denominator is x² - 4. The additional factor of the first fraction is x + 2: 1/(x - 2) = (x + 2)/(x² - 4); the second stays the same.
Class activity
“Denominator workshop”: pairs get cards with two or three fractions, find the common denominator and write the additional factors; another pair checks.
Practice
1
Find a common denominator of 1/(2x) and 5/(3y).
6xy
2
Bring 3/(a²b) to the denominator a²b²: what are the additional factor and the new numerator?
b; 3b
3
Find the common denominator of 2/(x + 1) and 3/(x² - 1) and the additional factor of the first fraction.
x² - 1; x - 1
4
Why does the sign change when 1/(b - a) is rewritten with the denominator a - b?
Because b - a = -(a - b); taking the minus out of the denominator puts it in front of the fraction: 1/(b - a) = -1/(a - b).