Question
Factorize the following expressions:
$x^3 + 6x^2 + 12x + 16$
$x^3 + 6x^2 + 12x + 16$
✓
Answer
$= x^3 + 6x^2 + 12x + 8 + 8$
$= x^3 + 3 \times x^2 \times 2 + 3 \times x \times 2^2 + 2^3 + 8$
$= (x + 2)^3 + 8$
$\big[\therefore$ $a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$$\big]$
$= (x + 2)^3 + 23 $
$= (x + 2 + 2)((x + 2)^2 - 2(x + 2) + 2^2)$
$\big[\therefore$ $a^3 + b^3 = (a + b)(a^2- ab + b^2)$$\big]$
$= (x + 2 + 2)(x^2 + 4 + 4x - 2x - 4 + 4)$
$\big[\therefore$ $(a + b)^2 = a^2 + b^2 + 2ab$$\big]$
$= (x + 4)(x^2 + 4 + 2x)$
$\therefore$ $x^3 + 6x^2 + 12x + 16 $
$= (x + 4)(x^2 + 4 + 2x)$
$= x^3 + 3 \times x^2 \times 2 + 3 \times x \times 2^2 + 2^3 + 8$
$= (x + 2)^3 + 8$
$\big[\therefore$ $a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$$\big]$
$= (x + 2)^3 + 23 $
$= (x + 2 + 2)((x + 2)^2 - 2(x + 2) + 2^2)$
$\big[\therefore$ $a^3 + b^3 = (a + b)(a^2- ab + b^2)$$\big]$
$= (x + 2 + 2)(x^2 + 4 + 4x - 2x - 4 + 4)$
$\big[\therefore$ $(a + b)^2 = a^2 + b^2 + 2ab$$\big]$
$= (x + 4)(x^2 + 4 + 2x)$
$\therefore$ $x^3 + 6x^2 + 12x + 16 $
$= (x + 4)(x^2 + 4 + 2x)$
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
In $\triangle\text{ABC},$ if $\angle\text{A}=40^\circ$ and $\angle\text{B}=60^\circ.$ Determine the longest and shortest sides of the triangle.
→ABC is a triangle in which $\angle\text{B}=2\angle\text{C}.$ D is a point on BC such that AD bisects $\angle\text{BAC}$ and $\text{AB}=\text{CD}.$ Prove that $\angle\text{BAC}=72^\circ.$
→Draw a line segment AB and by ruler and compasses, obtain a line segment of length $\frac{3}{4}\text{(AB)}.$
→Express $2.\overline{36}+0.\overline{23}$ as a fraction in simplest form.
→In figure, $\angle\text{AOC}$ and $\angle\text{BOC}$ from a linear pair. If a - 2b = 30°, find a and b.
→In figure, $\angle\text{AOF}$ and $\angle\text{FOG}$ from a linear pair. If $\angle\text{EOB}=\angle\text{FOC}=90^\circ$ and $\angle\text{DOC}=\angle\text{FOG}=\angle\text{AOB}=30^\circ.$
→
- Find the measure of $\angle\text{FOE},\angle\text{COB}$ and $\angle\text{DOE}.$
- Name all the right angles.
- Name three pairs of adjacent complementary angles.
- Name three pairs of adjacent supplementary angles.
- Name three pairs of adjacent angles
Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.
The students of Vidyalaya were asked to participate in a competition for making and decorating pen holders in the shape of a cylinder with a base, using cardboard. Each pen holder was to be of radius 3cm and height 10.5cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition?
The ratio between the radius of the base and height of a cylinder is $2 : 3$. If its volume is $1617cm^3$,
find the total surface area of the cylinder.
→find the total surface area of the cylinder.
Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following case:
$p(x) = x^3 - 6x^2 + 11x - 6, x = 1, 2, 3$
→$p(x) = x^3 - 6x^2 + 11x - 6, x = 1, 2, 3$