Find the rational roots of the polynomial $f(x)=2 x^{3}+x^{2}-7 x-6$
Given that $f(x)=2 x^{3}+x^{2}-7 x-6$
f(x) is a cubic polynomial with an integer coefficient . If the rational root in the form of p/q, the values of p are limited to factors of 6 which are ± 1, ± 2, ± 3, ± 6
and the values of q are limited to the highest degree coefficient i.e 2 which are ±1, ±2
here, the possible rational roots are ± 1, ± 2, ± 3, ± 6, ± 1/2, ± 3/2
Let, x = -1
$f(-1)=2(-1)^{3}+(-1)^{2}-7(-1)-6$
= - 2 + 1 + 7 - 6
= - 8 + 8
= 0
Let, x = 2
$f(-2)=2(2)^{3}+(2)^{2}-7(2)-6$
= (2 * 8) + 4 - 14 - 6
= 16 + 4 -14 - 6
= 20 - 20
= 0
Let, x = -3/2
$f(-3 / 2)=2(-3 / 2)^{3}+(-3 / 2)^{2}-7(-3 / 2)-6$
= 2(-27/8) + 9/4 - 7(-3/2) - 6
= (−27/4) + 9/4 - (-21/2) - 6
= – 6.75 + 2.25 + 10.5 - 6
= 12.75 - 12.75
= 0
But from all the factors only -1, 2 and −3/2 gives the result as zero
So, the rational roots of $2 x^{3}+x^{2}-7 x-6$ are $-1,2$ and $-3 / 2$
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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,