2.4. EXERCISES 45

and so you would need to solve the following system of equations

a+ f = 0,b−3 f + e = 0,d−4a+2 f −3e = 0c−4b−3d +2e = 0,4a−3c+2d = 0,4b+2c = 1

Solving the system of equations, a =− 34 ,b = 7

4 ,c =−3,d =−3, f = 34 ,e =

12 . It follows

that these two polynomials are relatively prime. Note how it was not necessary to factorthem to find out this information.

Example 2.3.2 Consider x−1 and x2−1. These are clearly not relatively prime. Considerthe above technique.

If they were relatively prime, then there would be a partial fractions expansion of theform 1

(x−1)(x2−1)= a

x−1 +bx+cx2−1 and so, multiplying gives

1 = a(x2−1

)+(bx+ c)(x−1)

= (a+b)x2 +(c−b)x− (a+ c)

Thus you would need to solve a+ b = 0,c− b = 0,−(a+ c) = 1. However, there is nosolution so we know these two are not relatively prime. Now again, I didn’t need to factorthese to draw this conclusion. Thus this gives another way to tell whether two polynomialsare relatively prime. It turns out that being able to do this is useful, as is shown later.

2.4 Exercises1. Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes

a nonzero number. Determine whether the given augmented matrix is consistent. Ifconsistent, is the solution unique?

■ ∗ ∗ ∗ ∗ | ∗0 ■ ∗ ∗ 0 | ∗0 0 ■ ∗ ∗ | ∗0 0 0 0 ■ | ∗

2. Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes

a nonzero number. Determine whether the given augmented matrix is consistent. Ifconsistent, is the solution unique?

■ ∗ ∗ ∗ ∗ | ∗0 ■ 0 ∗ 0 | ∗0 0 0 ■ ∗ | ∗0 0 0 0 ■ | ∗

3. Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes

a nonzero number. Determine whether the given augmented matrix is consistent. If