Name
October 28, 2013
Honors Advanced Mathematics
2.4 extra problems page 1
Greatest common factor and Least common multiple
The goal of these problems is to extend the concepts of greatest common factor (GCF) and
least common multiple (LCM) to polynomials. Answers may be left in factored form.
Greatest common factor
1. Using your prior knowledge of the concept of greatest common factor for numbers,
find GCF (42, 60). Show the method used to get your answer.
Definition of greatest common factor for polynomials
G(x) is called the greatest common factor (GCF) of polynomials P(x) and Q(x) if:
 G(x) is a factor of P(x),
 G(x) is a factor of Q(x),
 no higher degree polynomial is a factor of both P(x) and Q(x), and
 G(x) has a leading coefficient of 1 (that is, its highest degree term is of the form xn).
[The last requirement in the definition ensures that the GCF is unique. Without that stipulation,
once you had found a GCF, any constant multiple of it would also qualify as a GCF.]
2. Find GCF (x2 + 4x + 3, x2 + 2x – 3).
Hint: Factor both polynomials and see what the common factors are.
3. Find GCF (3x2 + 7x + 2, 3x3 + x2 – 3x – 1).
4. Find GCF (x3 – 2x2 – 9x + 18, x3 – 2x2 – 11x + 12).
5. Find GCF (x3 + 2x + 33, 2x3 – 5x2 + 19x + 11).
6. Find GCF (x3 + 9x2 + 24x + 16, 2x3 + 12x2 + 18x + 8).
7. Find GCF (x2 + 4x + 3, x2 – 4x + 3).
8. A pair of numbers or a pair of polynomials is said to be relatively prime if the pair has a
greatest common factor (GCF) of 1. Give examples of:
a. a pair of non-prime numbers that is relatively prime
b. a pair of non-prime numbers that isn’t relatively prime
c. a pair of cubic polynomials that is relatively prime
d. a pair of cubic polynomials that isn’t relatively prime
9. Suppose that P(x) and Q(x) are polynomials that can be factored completely into linear
factors, and that P(x) and Q(x) are relatively prime. What conclusion can be reached about
the zeros of polynomials P(x) and Q(x)?
10. Your answer to problem 1 should have been GCF(42, 60) = 6. However, if you regard 42 and
60 as being constant polynomials, and apply the GCF definition for polynomials, then
GCF(42, 60) can’t be 6 (because that would violate the last requirement of the definition).
So what is GCF(42, 60) when the numbers are regarded as polynomials?
Name
October 28, 2013
Honors Advanced Mathematics
2.4 extra problems page 2
Least common multiple
11. Using your prior knowledge of the concept of least common multiple for numbers,
find LCM (42, 60). Show the method used to get your answer.
Definition of least common multiple for polynomials
L(x) is called the least common multiple (LCM) of polynomials P(x) and Q(x) if:
 P(x) is a factor of L(x),
 Q(x) is a factor of L(x),
 no lower degree polynomial has factors of both P(x) and Q(x), and
 L(x) has a leading coefficient of 1 (that is, its highest degree term is of the form xn).
[Again the last requirement in the definition ensures that the LCM is unique.]
Note that problems 12–17 involve the same polynomials as problems 2–7. You can reuse any
factoring work that you did in those problems.
12. Find LCM (x2 + 4x + 3, x2 + 2x – 3).
13. Find LCM (3x2 + 7x + 2, 3x3 + x2 – 3x – 1).
14. Find LCM (x3 – 2x2 – 9x + 18, x3 – 2x2 – 11x + 12).
15. Find LCM (x3 + 2x + 33, 2x3 – 5x2 + 19x + 11).
16. Find LCM (x3 + 9x2 + 24x + 16, 2x3 + 12x2 + 18x + 8).
17. Find LCM (x2 + 4x + 3, x2 – 4x + 3).
18. a. Suppose that two numbers a and b are relatively prime (see problem 8).
Find LCM (a, b).
b. Suppose that two polynomials P(x) and Q(x) are relatively prime. Find LCM (P(x), Q(x)).
19. Suppose P(x) has degree p, Q(x) has degree q, and GCF(P(x), Q(x)) has degree g.
What is the degree of LCM(P(x), Q(x))?
Hint: The easiest case to think about is when P(x) and Q(x) both factor completely into
linear factors. Think about how many factors of each polynomial are shared with the other
polynomial, and how many factors are not shared.
20. Your answer to problem 11 should have been LCM(42, 60) = 420. However, if you regard 42
and 60 as being constant polynomials, and apply the LCM definition for polynomials, then
LCM (42, 60) can’t be 420 (because that would violate the last requirement of the definition).
So what is LCM (42, 60) when the numbers are regarded as polynomials?
21. Under what conditions is it true that an LCM of P(x) and Q(x) is their product P(x) · Q(x)?