Find LCD of Polynomials Using Factorization
Finding the LCD of polynomials using factorization is a Grade 7 algebra skill from Yoshiwara Intermediate Algebra. Students factor each polynomial denominator completely, then build the LCD as the product of all distinct factors raised to their highest powers, enabling addition and subtraction of rational expressions.
Key Concepts
Property The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator. To Find the LCD: 1. Factor each denominator completely. 2. Include each different factor in the LCD as many times as it occurs in any one of the given denominators.
Examples Find the LCD for $\frac{5}{6x^2y}$ and $\frac{7}{9xy^3}$. The factors are $2, 3^2, x^2, y^3$. The LCD is $2 \cdot 3^2 \cdot x^2 \cdot y^3 = 18x^2y^3$. Find the LCD for $\frac{x}{x^2 16}$ and $\frac{3x}{x^2+8x+16}$. The factored denominators are $(x 4)(x+4)$ and $(x+4)^2$. The LCD is $(x 4)(x+4)^2$. Find the LCD for $\frac{1}{5a(a 2)^3}$ and $\frac{b}{10a^2(a 2)}$. The factors are $2, 5, a^2, (a 2)^3$. The LCD is $10a^2(a 2)^3$.
Explanation The LCD is the smallest shared 'target' denominator for unlike fractions. Find it by factoring all denominators and taking the highest power of each unique factor that appears. This ensures all original denominators divide into it evenly.
Common Questions
How do you find the LCD of polynomial denominators?
Factor each denominator completely. The LCD is the product of all distinct factors, each taken to the highest power it appears in any denominator.
Why do you need to factor to find the LCD?
Factoring reveals common factors between denominators, so the LCD is as small as possible — you avoid over-multiplying.
What is the LCD of (x+1) and (x^2-1)?
Factor x^2-1 = (x+1)(x-1). The LCD is (x+1)(x-1).
How is finding LCD for polynomials similar to finding LCD for numbers?
Both require factoring (primes for numbers, polynomial factors for expressions) and then taking each factor at its maximum power.