First slide

Arithmetic progression

Question

If the roots of the equation $x^{3} - 12 x^{2} + 39 x - 28 = 0 are in AP,$ then their common difference will be

Easy

A

$\pm 1$
B

$\pm 2$
C

$\pm 3$
D

$\pm 4$

Solution

Since, the given equation is cubic, therefore we take three roots.
Let the roots be $a - d, a, a + d$
Sum of three numbers in AP = 3a = 12
$\Rightarrow a = 4 is a root.$
.'. The given equation $x^{3} - 12 x^{2} + 39 x - 28 = 0 can be$ re written as
$\begin{array}{ll} (x - 4) (x^{2} - 8 x + 7) = 0 \\ ∴ x = 1, 4, 7 or 7, 4, 1 \\ ∴ d = \pm 3 \end{array}$

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