Obtain all zeros of the polynomial $1(x)=x^{4}-3 x^{3}-x^{2}+9 x-6$, if two of its zeros are $-\sqrt{3}$ and $\sqrt{3}$
We know that if $x=\alpha$ is a zero of a polynomial, then $(x-\alpha)$ is a factor of $f(x)$.
Since $-\sqrt{3}$ and $\sqrt{3}$ are zeros of $f(x)$.
Therefore
$(x+\sqrt{3})(x-\sqrt{3})=x^{2}+\sqrt{3} x-\sqrt{3} x-3$
$=x^{2}-3$
$x^{2}-3$ is a factor of $f(x)$. Now, we divide $f(x)=x^{4}-3 x^{3}-x^{2}+9 x-6$ by $g(x)=x^{2}-3$ to find the other zeros of $f(x)$.
By using division algorithm we have, $f(x)=g(x) \times q(x)+r(x)$.
$x^{4}-3 x^{3}-x^{2}+9 x-6=\left(x^{2}-3\right)\left(x^{2}-3 x+2\right)+0$
$x^{4}-3 x^{3}-x^{2}+9 x-6=\left(x^{2}-3\right)\left(x^{2}-2 x-1 x+2\right)$
$x^{4}-3 x^{3}-x^{2}+9 x-6=\left(x^{2}-3\right)[x(x-2)-1(x-2)]$
$x^{4}-3 x^{3}-x^{2}+9 x-6=\left(x^{2}-3\right)[(x-1)(x-2)]$
$x^{4}-3 x^{3}-x^{2}+9 x-6=(x-\sqrt{3})(x+\sqrt{3})(x-1)(x-2)$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,