Ask a Doubt
Request a call back

Join NOW to get access to exclusive study material for best results

CBSE Class 10 Answered

Find k so that X2+2x+k is a factor of 2x4+x3-14x2+5x+6.Also find the zeroes of the two polynomials.

Asked by bosesreyan | 02 May, 2015, 09:18: PM
answered-by-expert Expert Answer
Use long division method for the given polynomials as shown below:
 
space space space space space space space space space space space space space space space space space space 2 x squared minus 3 x minus 8 minus 2 k x squared plus 2 x plus k long division enclose 2 x to the power of 4 plus x cubed minus 14 x squared plus 5 x plus 6 end enclose space space space space space space space space space space space space space space space space space space space 2 x to the power of 4 plus 4 x cubed plus 2 k x squared space space space space space space space space space space space space space space space space space space bottom enclose negative space space space space space minus space space space space space minus space space space space space space space space space space space space space space space space space space space end enclose space space space space space space space space space space space space space space space space space space space space space space space space space minus 3 x cubed minus 14 x squared minus 2 k x squared plus 5 x plus 6 space space space space space space space space space space space space space space space space space space space space space space space space space minus 3 x cubed minus 6 x squared space space space space space space space space space space space space space minus 3 k x space space space space space space space space space space space space space space space space space space space space space space space space space space bottom enclose plus space space space space space space plus space space space space space space space space space space space space space space space space space space space plus space space space space space space space space space space end enclose space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus 8 x squared minus 2 k x squared plus 5 x plus 3 k x plus 6 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus 8 x squared minus 2 k x squared minus 16 x minus 4 k x minus 8 k minus 2 k squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space bottom enclose plus space space space space space space space plus space space space space space space space space plus space space space space space space plus space space space space space space space plus space space space space plus space space space space space space end enclose space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses 21 plus 7 k close parentheses x plus 2 k squared plus 8 k plus 6 space space space space space space space space space space space space space space space space space space space space space space
We get the remainder as open parentheses 21 plus 7 k close parentheses x plus 2 k squared plus 8 k plus 6.
Since, x squared plus 2 x plus k is a factor of the given polynomial, the remainder should be zero.
Hence,
21+7k=0 and 2k2+8k+6=0 at the same time.
k= -3 satisfies both equations. Hence, k=-3.
So we can write the polynomial as
2 x to the power of 4 plus x cubed minus 14 x squared plus 5 x plus 6 equals open parentheses x squared plus 2 x minus 3 close parentheses open parentheses 2 x squared minus 3 x minus 2 close parentheses
Hence, the zeroes of x squared plus 2 x minus 3 are
x squared plus 2 x minus 3 equals 0 rightwards double arrow x squared plus 3 x minus x minus 3 equals 0 rightwards double arrow x open parentheses x plus 3 close parentheses minus 1 open parentheses x plus 3 close parentheses equals 0 rightwards double arrow open parentheses x plus 3 close parentheses open parentheses x minus 1 close parentheses equals 0 rightwards double arrow x equals negative 3 space o r space x equals 1
The above two are factors of the 4th degree polynomial as well.
The other two roots of the 4th degree polynomial are roots of the quadratic
2 x squared minus 3 x minus 2 equals 0 rightwards double arrow 2 x squared minus 4 x plus x minus 2 equals 0 rightwards double arrow 2 x open parentheses x minus 2 close parentheses plus 1 open parentheses x minus 2 close parentheses equals 0 rightwards double arrow open parentheses x minus 2 close parentheses open parentheses 2 x plus 1 close parentheses equals 0 rightwards double arrow x equals 2 space o r space x equals negative 1 half
Answered by satyajit samal | 04 May, 2015, 12:50: PM

Application Videos

Concept Videos

Multiple Choice Questions Practice Test
Answered Unanswered My Questions My Answers
CBSE 10 - Maths
What should be added to the polynomial x^3 - 2x^2 -3x -4 so, that it is completely divisible by x^2 - x
Asked by mv716304 | 01 Apr, 2022, 05:16: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
x^2+2√2x-6
Asked by mahekk101 | 14 Sep, 2021, 08:24: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
if the two zeros of quadratic polynomial 7x²-15x-k are reciprocals of each other the value of k?
question image
Asked by bhavikabhatia1125 | 03 Aug, 2021, 10:29: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
If the polynomial 2x⁴ + x³ - 3x² +18x + 24 is divided by another polynomial x² + 3x - m and  the remainder comes out to be 4x + n. Find the value of m and n respectively.
Asked by Aasthaachd | 08 Jun, 2021, 01:39: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
Divide x^4 -5x+6 by x^2 -2
Asked by rk4353620 | 05 Jun, 2021, 01:52: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
Find whether 3x+2 is a factor of 3x^4+ 5x^3+ 13x-x^2 + 10
Asked by diyayadav1441 | 27 Jul, 2020, 03:59: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
If two of the zeroes of the polynomial f(x)=x4-4x3-20x2+104x-105 are 3+√2 and 3-√2,then use the division algorithm to find the other zeroes of f(x).
Asked by saiprathumnan35 | 26 Jun, 2020, 07:17: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
2x⁴+x³-14x²-15x-8 divide by x²+3x+2
Asked by Gurnoorsingh0852 | 03 Jun, 2020, 03:54: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
Using division algorithm find the quotient and remainder of the following: I) x³-6x²+11x-6 by x²+x+1
Asked by renuneeraj2005 | 02 Jun, 2020, 11:00: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
CBSE 10 - Maths
if m and n are divided by 4 then 2 and 3 are respectively the remainders find the remainder when m square + n square is divided by 4
question image
Asked by dash.ak35 | 08 Apr, 2020, 01:19: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
My Class Videos Tests ₹ Plans Ask a Doubt
Get Latest Study Material for Academic year 24-25 Click here
×