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  • Find k so that x^2 + 2x + k is a factor of 2x^4 + x3 – 14x^2 + 5x + 6. Also find all the zeroes of the two polynomials.

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.

Answers (1)

The given polynomial is 2x^4 + x^3 - 14x^2 + 5x + 6

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

Use division algorithm

Here, x(7k + 21) + (6 + 8k + 2k^2) is the remainder

It is given that x^2 + 2x + k is a factor hence remainder = 0

x(7k + 21) + (6 + 8k + 2k^2) = 0.x + 0

By comparing L.H.S. and R.H.S.

7k + 21 = 0

k = – 3                         ….(1)

2k^2 + 8k + 6 = 0

2k^2 + 2k + 6k + 6 = 0

2k(k + 1) + 6(k + 1) = 0

(k + 1) (2k + 6) = 0

k = –1, –3                    …..(2)

From (1) and (2)

k = –3

Hence, 2x^4 + x^3 - 14x^2 + 5x + 6 = (x^2 + 2x + k) (2x^2 - 3x - 8 - 2k)

Put k = –3

= (x^2 + 2x - 3) (2x^2 - 3x - 8 + 6)

= (x^2 + 2x - 3) (2x^2 - 3x - 2)

= (x^2 + 3x - x - 3) (2x^2 - 4x + x - 2)

= (x(x + 3) - 1(x + 3)) (2x(x - 2) + 1(x - 2))

= (x + 3)(x -1)(x - 2) (2x + 1)

x + 3 = 0                      x – 1 = 0                      x – 2 = 0                      2x + 1 = 0

x = –3                          x = 1                            x = 2                            2x = – 1

                                                                                                             x=\frac{-1}{2}

Here zeroes of  x^2 + 2x - 3is –3, 1

Zeroes of 2x^2 - 3x - 2 is 2, \frac{-1}{2} .

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