Question
Using factor theorem, factorize the following polynomials:
$x^3 - 23x^2 + 142x - 120$
$x^3 - 23x^2 + 142x - 120$
✓
Answer
Let $p(x) = x^3 - 23x^2 + 142x - 120$
We shall now look for all the factors of -120. Some of these are $1,\pm2,\pm3,\pm,\pm4,\pm5,\pm6,\pm8\\\pm12,\pm15,\pm20,\pm24,\pm30,\pm,60$
By trial, we find that $p(1) = 0.$
So $x - 1 $is a factor of $p(x).$
Now, we see that $x^3 - 23x^2 + 142x - 120 = x^3 - x^2- 22x^2 + 22x +120x - 120$
$= x^2(x - 1) - 22x(x - 1) + 120(x - 1)$
$= (x - 1)(x^2 - 22x + 120)$ [Taking (x - 1) common]
We could have also got this by dividing p(x) by $x - 1.$
Now $x^2 - 22x + 120$ can be factorised either by splitting the middle term or by using the factor theorem.
By splitting the middle term, we have:
$x^2 - 22x + 120 = x^2 - 12x - 10x+ 120$
$= x(x - 12) - 10(x - 12)$
$= (x - 12)(x - 10)$
So,
$x^3 - 23x^2 + 142x - 120 = (x - 1)(x - 10)(x - 12)$
We shall now look for all the factors of -120. Some of these are $1,\pm2,\pm3,\pm,\pm4,\pm5,\pm6,\pm8\\\pm12,\pm15,\pm20,\pm24,\pm30,\pm,60$
By trial, we find that $p(1) = 0.$
So $x - 1 $is a factor of $p(x).$
Now, we see that $x^3 - 23x^2 + 142x - 120 = x^3 - x^2- 22x^2 + 22x +120x - 120$
$= x^2(x - 1) - 22x(x - 1) + 120(x - 1)$
$= (x - 1)(x^2 - 22x + 120)$ [Taking (x - 1) common]
We could have also got this by dividing p(x) by $x - 1.$
Now $x^2 - 22x + 120$ can be factorised either by splitting the middle term or by using the factor theorem.
By splitting the middle term, we have:
$x^2 - 22x + 120 = x^2 - 12x - 10x+ 120$
$= x(x - 12) - 10(x - 12)$
$= (x - 12)(x - 10)$
So,
$x^3 - 23x^2 + 142x - 120 = (x - 1)(x - 10)(x - 12)$
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 Fig. $X$ and $Y$ are the mid-points of $AC$ and $AB$ respectively, $QP || BC$ and $CYQ$ and $BXP$ are straight lines. Prove that $ar\ (ABP) = ar\ (ACQ).$
→The inner diameter of a cylindrical wooden pipe is 24cm and its outer diameter is 28cm. The length of the pipe is 35cm. Find the mass of the pipe, if 1cm3 of wood has a mass of 0.6g
→$\Big[$Hint: Assume $\pi=\frac{22}{7},$ unless stated otherwise$\Big]$
If the mean of $x + 2, 2x + 3, 3x + 4, 4x + 5 is x + 2$, find $x$.
→→
ABCD is a rhombus. Show that diagonal AC bisects $\angle\text{A}$ as well as $\angle\text{C}$ and diagonal BD bisects $\angle\text{B}$ as well as $\angle\text{D}.$
→If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that the quadrilateral so formed is cyclic.
Prove that $(a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b )(b + c)(c + a).$
→Three coins are tossed simultaneously $200$ times with the following frequencies of different outcomes:
Find the probability of getting at most two heads.
→|
Outcome
|
$3$ heads
|
$2$ heads
|
$1$ head
|
No head
|
|
Frequency
|
$23$
|
$72$
|
$77$
|
$28$
|
A square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.
A ship sails out to an island at the rate of $15\ km/hr$ and sails back to the starting point at $10\ km/hr$. Find the average sailing speed for the whole journey.
→