Example Card: Finding the LCM of Polynomials
Find the Least Common Multiple of polynomials in Grade 9 algebra by factoring each polynomial completely, then taking the highest power of every distinct factor for use as the LCD in rational expressions.
Key Concepts
Factoring polynomials is the secret to finding their LCM. Let's try this with the second key idea from this lesson, identifying the LCM of polynomials.
Example Problem Find the LCM of $(8x^2+24x)$ and $(10x+30)$.
Step by Step 1. First, factor each binomial to find its prime components. 2. The Greatest Common Factor (GCF) of the terms in $(8x^2+24x)$ is $8x$. Factor it out. $$ (8x^2+24x) = 8x(x+3) = 2 \cdot 2 \cdot 2 \cdot x(x+3) $$ 3. The GCF of the terms in $(10x+30)$ is $10$. Factor it out. $$ (10x+30) = 10(x+3) = 2 \cdot 5(x+3) $$ 4. Now, identify all the unique factors. We have the numeric factors 2 and 5, the variable factor $x$, and the binomial factor $(x+3)$. 5. To build the LCM, take the greatest number of times each factor appears in any single expression. The factor 2 appears most (three times). The factor 5 appears once. The factor $x$ appears once. The factor $(x+3)$ appears once in each. $$ \operatorname{LCM} = 2 \cdot 2 \cdot 2 \cdot 5 \cdot x(x+3) $$ 6. Multiply these factors to get the final LCM. $$ \operatorname{LCM} = 40x(x+3) $$ The LCM is $40x(x+3)$.
Common Questions
How do you find the LCM of polynomial expressions?
Factor each polynomial completely. List all distinct factors. For each factor, take the highest power that appears in any of the polynomials. Multiply these together to form the LCM.
What is the LCM of x² - 4 and x + 2?
Factor x² - 4 = (x+2)(x-2). The second polynomial is already (x+2). The distinct factors are (x+2) and (x-2), each to the first power. LCM = (x+2)(x-2) = x² - 4.
Why do you need the LCM when adding or subtracting rational expressions?
To add or subtract fractions with different denominators, you need a common denominator. The LCM of the denominators is the Least Common Denominator (LCD), which keeps the resulting fraction in simplest form.