Question
Using factor theorem, factorize the following polynomials: $x^3 + 6x^2 + 11x + 6$
✓
Answer
Let x = 1 $\text{f(1)}=1^3+6(2)^2+11(1)+6\neq0$
Let $x = -1 f(-1) = (-1)^3 + 6(-1)^2 + 11(-1) + 6 = 12 - 12 = 0$
$\therefore$ $x = -1$ is a solution
$\Rightarrow x + 1 = 0 i. e (x + 1)$ is a factor of $f(x)$
By division algorithm $x^3 + 6x^2 + 11x + 6$
$= (x + 1)(x^2 + 5x + 6)$
$= (x + 1)(x^2 + 2x + 3x + 6)$
$= (x + 1)(x(x + 2) + 3(x + 2))$
$= (x + 1)(x + 2)(x + 3)$
$\therefore x^3 + 6x^2 + 11x + 6$
$=(x + 1)(x + 2)(x + 3)$
Let $x = -1 f(-1) = (-1)^3 + 6(-1)^2 + 11(-1) + 6 = 12 - 12 = 0$
$\therefore$ $x = -1$ is a solution
$\Rightarrow x + 1 = 0 i. e (x + 1)$ is a factor of $f(x)$
By division algorithm $x^3 + 6x^2 + 11x + 6$
$= (x + 1)(x^2 + 5x + 6)$
$= (x + 1)(x^2 + 2x + 3x + 6)$
$= (x + 1)(x(x + 2) + 3(x + 2))$
$= (x + 1)(x + 2)(x + 3)$
$\therefore x^3 + 6x^2 + 11x + 6$
$=(x + 1)(x + 2)(x + 3)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Three coins were tossed $30$ times. Each time the number of heads occurring was noted down as follows:
→$0, 1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 2, 2, 0, 1, 2, 1, 3, 0, 1, 1, 2, 3, 2, 2, 0$
Prepare a frequency distribution table for the data given above.It costs $₹ 3300$ to paint the inner curved surface of a cylindrical vessel $10\ m$ deep at the rate of ₹ $30$ per $m^2$. Find the:
$i.$ Inner curved surface area of the vessel.
$ii.$ Inner radius of the base.
$iii.$ Capacity of the vessel.
→$i.$ Inner curved surface area of the vessel.
$ii.$ Inner radius of the base.
$iii.$ Capacity of the vessel.
Plot the following points on the graph paper:$(0, 7)$
→In Figure, if a is greater than b by one third of a right angle. Find the value of $a$ and $b.$
→The mean of the following data is $20.6$ Find the value of $p$.
→|
x:
|
$10$
|
$15$
|
$p$
|
$25$
|
$35$
|
|
f:
|
$3$
|
$10$
|
$25$
|
$7$
|
$5$
|
Hameed has built a cubical water tank with lid for his house, with each other edge $1.5\ m$ long. He gets the outer surface of the tank excluding the base, covered with square tiles of side $25\ cm$ . Find how much he would spend for the tiles if the cost of tiles is $₹ 360$ per dozen.
→$ABCD$ is a trapezium in which $AB || DC, DC = 30\ cm$ and $AB = 50\ cm.$ If $X$ and $Y$ are, respectively the mid-points of $AD$ and $BC,$ prove that $\text{ar}(\text{DCYX})=\frac{7}{9}\ \text{ar}(\text{XYBA})$
→A sphere of radius $5\ cm$ is immersed in water filled in a cylinder, the level of water rises $\frac{5}{3}$cm. Find the radius of the cylinder.
→Two cones have their heights in the ratio $1 : 3$ and the radii of their bases in the ratio $3 : 1$. Find the ratio of their volumes.
→ If $\text{x}=2+\sqrt{3},$ find the value of $\text{x}^3+\frac{1}{\text{x}^3}.$
→