Question
Find a cubic polynomial whose zeroes are $\frac{1}{2},$ 1 and −3.
✓
Answer
If the zeroes of the cubic polynomial are a, b and c then the cubic polynomial can be found as:
$x^3- (a + b + c)x^2 + (ab + bc + ca)x - abc ...(1)$
Let $\text{a}=\frac{1}{2},$ b = 1 and c = -3
Substituting the values in (1), we get
$\text{x}^3-\Big(\frac{1}{2}+1-3\Big)\text{x}^2+\Big(\frac{1}{2}-3-\frac{3}{2}\Big)\text{x}-\Big(\frac{-3}{2}\Big)$
$\Rightarrow\text{x}^3-\Big(\frac{-3}{2}\Big)\text{x}^2-\text{4x}+\frac{3}{2}$
$\Rightarrow\text{2x}^3+\text{3x}^2-\text{8x}+3$
$x^3- (a + b + c)x^2 + (ab + bc + ca)x - abc ...(1)$
Let $\text{a}=\frac{1}{2},$ b = 1 and c = -3
Substituting the values in (1), we get
$\text{x}^3-\Big(\frac{1}{2}+1-3\Big)\text{x}^2+\Big(\frac{1}{2}-3-\frac{3}{2}\Big)\text{x}-\Big(\frac{-3}{2}\Big)$
$\Rightarrow\text{x}^3-\Big(\frac{-3}{2}\Big)\text{x}^2-\text{4x}+\frac{3}{2}$
$\Rightarrow\text{2x}^3+\text{3x}^2-\text{8x}+3$
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
The angle of elevation of an aeroplane from a point on the ground is 45°. After flying for 15 seconds, the elevation changes to 30°. If the aeroplane is flying at a height of 2500 metres, find the speed of the airoplane.
Solve the following quadratic equation:
$3\Big(\frac{\text{3x}-1}{\text{2x}+3}\Big)-2\Big(\frac{\text{2x}+3}{\text{3x}-1}\Big)=5$ $\text{x}\neq\frac{1}{3},\ \frac{-3}{2}$
→$3\Big(\frac{\text{3x}-1}{\text{2x}+3}\Big)-2\Big(\frac{\text{2x}+3}{\text{3x}-1}\Big)=5$ $\text{x}\neq\frac{1}{3},\ \frac{-3}{2}$
Find the area of quad. ABCD in which AB = 42cm, BC = 21cm, CD = 29cm, DA = 34cm and diag. BD = 20cm.
Find the area of the shaded region in the given figure, where a circular arc of radius 6cm has been drawn with vertex of an equilateral triangle of side 12cm as centre and a sector of circle of radius 6cm with centre B is made. $\big[\text{Use }\sqrt{3}=1.73\text{ and }\pi=3.14\big]$
→An iron pillar has some part in the form of a right circular cylinder and remaining in the form of a right circular cone.The radius of the base of each of cone and cylinder is $8 \ cm.$ The cylindrical part is $240 \ cm$ high and the conical part is $36 \ cm$ high. Find the weight of the pillar if one cubic $\ cm$ of iron weighs $7.8$ grams.
→300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of students.
A metallic cylinder has radius 3cm and height 5cm. To reuce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of $\frac{3}{2}\text{cm}$ and its depth is $\frac{8}{9}\text{cm}.$ Calculate the ratio of the volume of metal left in the cylinder to the volume of metal taken out in conical shape.
→For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
$-2\sqrt{3},-9$
→$-2\sqrt{3},-9$
The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.
A box contains cards numbered 3, 5, 7, 9, ..., 35, 37. A card is drawn at random from the box. Find the probability that the number on the card is a prime number.