find k so that x2+2x+k is a factor of 2x4+x3-14x2+5x+6.also,find all zeroes of the polynomial.

Asked by FEHMEEDAH NAZNEEN | 31st Aug, 2010, 09:34: AM

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  | 5th May, 2015, 12:04: PM

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