# The polynomial #8x^3-36x^2+22x+21=0# has roots which form an arithmetic progression. How do you find the roots?

##### 2 Answers

Use the information that the roots are in arithmetic relation to reduce the order of the problem from cubic to quadratic

#### Explanation:

Firstly, let's write our equation with "monic" first term, i.e. with first coefficient equal to 1.

We know from the fact that it has highest polynomial term cubic that there are three roots. We know from given info that the three roots are separated by the same number - write them as

Multiply out the two symmetric brackets:

(Note that the chosen symmetric way of writing the roots has made this a simpler expression that it would otherwise have been - it's always worth keeping your algebra as tidy as possible - less confusing, harder to make mistakes)

Multiply out the third bracket:

Now we can immediately deduce

In the third coefficient we have:

So we have two solutions for

We now double-check our answer by substituting it back in to the original equation (this is important to do! I caught a mistake in my working by doing it...):

#### Explanation:

We have:

# 8x^3-36x^2+22x+21 = 0 #

We are given that the roots are in Arithmetic Progression. Let us denote the roots by

# x_1 = alpha #

# x_2 = beta = alpha+d #

# x_2 = gamma = alpha+2d #

The sum of roots properties gives us:

# alpha + beta + gamma = -b/a #

# :. (alpha ) + (alpha + d) + (alpha+2d) = -(-36/8) #

# :. 3alpha +3d = 9/2 #

# :. alpha +d = 3/2 # ..... [A]

The product of roots properties gives us:

# alpha \ beta \ gamma = -d/a #

# :. (alpha)(alpha+d)(alpha+2d) = -21/8 #

# :. alpha(3/2)(alpha+2(3/2-alpha)) = -21/8 # (using [A])

# :. 3/2 alpha(alpha+3-2alpha) = -21/8 #

# :. 12 alpha(3-alpha) = -21 #

# :. 36 alpha-12alpha^2=-21 #

# :. 12alpha^2 -36alpha-21 = 0 #

# alpha^2 -36/12alpha-21/12 = 0 #

# :. (alpha -3/2)^2-(3/2)^2-21/12 = 0 #

# :. (alpha -3/2)^2 = 21/12 +9/4 #

# :. alpha -3/2 = +-sqrt(4) #

# :. alpha = 3/2+-2 #

# :. alpha = -1/2, 7/2 #

And, with each of these solutions we get (using [A]):

**Case (1):**

# alpha = -1/2 => -1/2+d=3/2 => d = 2#

Leading to the roots:

# (alpha ),(alpha + d),(alpha+2d) # ie#-1/2, 3/2, 7/2#

**Case (2):**

# alpha = \ \ \ \ \ 7/2 => \ \ \ \ \ 7/2+d=3/2 => d = -2#

Leading to the roots:

# (alpha ),(alpha + d),(alpha+2d) # ie#7/2, 3/2, -1/2#

So, in both cases we have the roots:

#-1/2, 3/2, 7/2#