Question

Find k so that x^2 + 2x + k is a factor of 2x^4 + x^3 - 14x^2 + 5x + 6 also find all the zeroes of the two polynomials.


Class 10



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Answer 1

HELLO DEAR,

GIVEN:-

x^2 + 2x + k is a factor of 2x^4 + x^3 - 14x^2 + 5x + 6

so,

x² + 2x + k )2x⁴ + x³ - 14x² + 5x + 6(2x² - 3x - (8 +2k)

                2x⁴ + 4x³ + 2kx²

               =======================

         -3x³ - x²(14 + 2k) + 5x

         -3x³ - 6x² - 3kx

       ===========================                    

        -x²(8 + 2k) + x(5 + 3k) + 6          

        -x²(8 + 2k) - 2x(8 + 2k) - k(8 + 2k)

       =============================

            x(21 + 7k) + (2k² + 8k + 6)

now,

given that:-

x^2 + 2x + k is a factor of 2x^4 + x^3 - 14x^2 + 5x + 6

so, remainders should be 0.

thus, x(21 + 7k) + (2k² + 8k + 6) = 0

comparing cofficient of x both side,

(21 + 7k) = 0

7k = -21

k = -3,

hence, x² + 2x + k = x² + 2x - 3

roots of x² + 2x - 3 = 0 is

x² + 3x - x - 3 = 0

x(x + 3) - 1(x + 3) = 0

(x - 1)(x + 3) = 0

x = 1 , x = -3

AND,

roots of 2x² - 3x - (8 +2k) = 2x² - 3x - (8 - 6)

$\Rightarrow$ 2x² - 3x - 2 = 0

2x² - 4x + x - 2 = 0

2x(x - 2) + 1(x - 2) = 0

(2x + 1)(x - 2) = 0

x = 2 , x = -1/2

I HOPE ITS HELP YOU DEAR,

THANKS