Name: Introduction to Algebra
Uploaded: 2019-07-19
Duration: 11 min 18 s
Description: In this video, we have discussed basic concepts of Algebra. Having a clear idea of the concept of this chapter will helps you to solve problems which are related to this topic.

Question

It is given that

(x + a)

is a factor of

(x^{3} + 4 x^{2} + 7 a x + 4 a)

. Find the possible values of

a

.

Accepted Answer

Let $f (x) = (x^{3} + 4 x^{2} + 7 a x + 4 a)$ ,

Given that, $(x + a)$ is factor of $f (x)$ .

We know that, if $(x - a)$ is a factor of $f (x)$ then, $f (a) = 0$ .

From factor theorem, $f (- a) = 0$

$\Rightarrow {(- a)}^{3} + 4 {(- a)}^{2} + 7 a (- a) + 4 a = 0 \Rightarrow - a^{3} + 4 a^{2} - 7 a^{2} + 4 a = 0 \Rightarrow - a^{3} - 3 a^{2} + 4 a = 0 \Rightarrow a^{3} + 3 a^{2} - 4 a = 0 . . . . . . . . (i) e q .$

Let, $g (a) = a^{3} + 3 a^{2} - 4 a$

$\Rightarrow g (1) = 1^{3} + 3 {(1)}^{2} - 4 (1) = 0$

$(a - 1)$ is the factor of $g (a)$ .

$a^{3} + 3 a^{2} - 4 a$ can be written as,

$\Rightarrow a^{3} + 3 a^{2} - 4 a = (a - 1) (p a^{2} + q a + r) \Rightarrow a^{3} + 3 a^{2} - 4 a = p a^{3} + q a^{2} - p a^{2} + a r - q a - r \Rightarrow a^{3} + 3 a^{2} - 4 a = p a^{3} + (q - p) a^{2} + a (r - q) - r$

Comparing coefficient of $a^{3}$ is,

$\Rightarrow p = 1$

Comparing constant,

$\Rightarrow r = 0$

Comparing coefficient of $a^{2}$ is,

$\Rightarrow q - p = 3 \Rightarrow q - 1 = 3 \Rightarrow q = 4$

Hence, $g (a) = (a - 1) (a^{2} + 4 a + 0)$

$\Rightarrow g (a) = (a - 1) (a^{2} + 4 a) \Rightarrow g (a) = a (a - 1) (a + 4)$

From $(i) e q .$ ,

$a (a - 1) (a + 4) = 0 \Rightarrow a = 0, 1, - 4$

It is given that $(x + a)$ is a factor of $(x^{3} + 4 x^{2} + 7 a x + 4 a)$ . Find the possible values of $a$ .

Important Questions on Algebra

Learn and Prepare for any exam you want !

It is given that x+a is a factor of x3+4x2+7ax+4a. Find the possible values of a.

Important Questions on Algebra

Learn and Prepare for any exam you want !

It is given that $(x + a)$ is a factor of $(x^{3} + 4 x^{2} + 7 a x + 4 a)$ . Find the possible values of $a$ .