Teaching Notes 6.7: Finding the Least Common Multiple
of Polynomials
The concept of finding the least common multiple of polynomials is essential for finding the least
common denominator of algebraic fractions. Errors occur when students factor a polynomial
incorrectly or when they do not consider all of the common factors.- Explain the process for finding the least common multiple (LCM) of whole numbers to your
students. Present this example: Find the LCM of 20 and 75. Note that students are to find the
prime factorization of each number and express the prime factorization using exponents.
20 = 22 × 5 ; 75 = 52 ×3 Remind your students that the LCM is expressed as the product
of each base raised to the highest power of each factor. Therefore, the LCM of 20 and 75 is
22 × 52 ×3 or 300. You might find it helpful to review 1.10: ‘‘Writing Prime Factorization.’’ - Explain that the process of finding the LCM of algebraic fractions follows the same procedure.
But instead of factoring numbers, students will be factoring polynomials. - Review the information and example on the worksheet with your students. Be sure that your
students understand each step of the example.
EXTRA HELP:
Use all of the factors, not just the ones that are common to both polynomials when writing the LCM.ANSWER KEY:
(1)(x−2)(x−3)(x−4) (2)(x−2)(x+4)(x−4) (3)(x−5)^2 (x+4) (4)(x−1)(x+4)(x+2)(5)(x+4)^2 (x+1) (6)x^2 (x+1)(x+6) (7)5(x^2 +10) (8)(x−1)(x+4)(x+8)(x+2)
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(Challenge)Deena is correct. If two polynomials (or whole numbers) have no common factors
other than 1, the LCM is the product of the factors.
------------------------------------------------------------------------------------------236 THE ALGEBRA TEACHER’S GUIDE